In the very first episode, there's a rather understated one between O'Brien and Captain Picard. During the episode, O'Brien transfers from the Enterprise to DS9 as part of a promotion. When the time finally comes to leave, he stops by the captain's ready room on the bridge, but decides against interrupting when told that Picard is busy at the moment. But when he heads to the transporter room, Picard walks in right after after being told that O'Brien stopped by to see him.
Picard: This is your favourite transporter room, isn't it?
O'Brien: Number three, yes sir.
Picard: You know, yesterday I called down here and I asked for you without thinking . . . it won't be quite the same.
O'Brien: It's just a transporter room, sir.
The whole scene is very understated but also meaningful. Even the comment about it being "just a transporter" is perfectly in O'Brien's character, pretty much reassuring the captain that it'll be just fine without him to look after things. And O'Brien gets the perfect sendoff from his ST:TNG career when Picard himself takes over the transporter controls and personally beams O'Brien away to his new station.
The pilot earns its Heartwarming stripes by the mere Character DevelopmentThe Captain, okay really Commander, undergoes in the space of one episode. When we first meet Sisko, he's literally consumed with rage and regret over the death of his wife, going so far as to come this close to outright insubordination to Captain Picard, such is his contempt for the man he holds responsible for his wife's death. He doesn't want to be in Starfleet and he doesn't want to be on a wrecked Cardassian station. But, a run-in with some non-corporeal aliens and Sisko rediscovers that the end of one thing is an opportunity for something different; maybe something greater...this is one of Star Trek's founding themes. The revitalized Commander asks Picard to rescind his request for a transfer and makes it clear he's at DS9 to stay. Picard simply smiles and wishes Sisko good luck before shaking hands and leaving.
This line from the beginning of "The Adversary"...
Jake: "Dad, there's something I've been wanting to say to you for a long time and now that I finally have the chance, I'm going to make it short and simple. Congratulations...Captain Sisko!"
The moment in "The Way of the Warrior" where Worf and O'Brien reminisce about the events surrounding "The Best of Both Worlds". It becomes a Tear Jerker when they start eulogizing the Enterprise-D because, in a meta sense, they are also eulogizing the Next Generation.
Worf: We were like warriors from the ancient sagas. There was nothing we could not do.
The episode "In the Cards" is one of the franchise's few truly successful comedy episodes, focusing on Jake's attempt to cheer up his father in the wake of the looming war by getting him a mint condition Willie Mays rookie card. This ultimately involves both him and Nog having to do favors for a bunch of other people: do jobs for O'Brien so he can go kayaking on the holodeck, retrieve Bashir's childhood teddy bear after his ex-girlfriend stole it, punch up an important speech for Kira, and fix distortions in Worf's collection of opera recordings. At the episode's end, we see the results of all this as Captain Sisko records a log entry that somehow people on the station seem happy again. Finally, he receives the hard-won card from his son and wraps him in a huge hug.
The end of "The Assignment", after O'Brien saves his wife Keiko from a disembodied alien that had possessed her in order to blackmail O'Brien into sabotaging the station in an effort to destroy the Wormhole. Lovely exchange between those two at the end;
Miles: You don't have to talk about it if you don't want to.
Keiko: It helps to talk about it. Besides, I never want to forget how you fought for me.
Also from "The Assignment" there was the subplot about Rom joining the station engineering crew. After showing unwavering loyalty to O'Brien in his struggle to save Keiko, even keeping the faith when he was arrested for sabotage and O'Brien was forced to let him take the blame, he is rewarded at the end when O'Brien promotes him to the day shift in gratitude. It's quite genuinely endearing.
And while we're talking about Miles and Keiko, there's a sweet scene from the second season ep "Rivals", in which O'Brien is psyching himself up for a racquetball match against Dr Bashir, who had beaten him soundly earlier. Keiko stands beside him, assuming the attitude of a samurai's wife as her husband prepares for battle, while reminding him, "Remember, Miles. Win or lose, tonight we celebrate." Before he leaves, Keiko presents Miles with a Japanese silk scarf scented with her perfume. She wraps the scarf around his head, kisses him and says, "Kick his butt."
Then there was when Keiko and Molly returned from a year-long botanical survey on Bajor. Miles, used to spending his time with Julian (to compensate for Keiko and Molly being absent) doing things such as re-enacting the Battle of Britain, finds the transition surprisingly difficult and even goes so far to try to teach Molly how to play darts - something he always did with Julian. As much as it pains him, he is determined to spend time with his family and be a good husband and father. Keiko, understanding his pain, concocts a little story about how depressed Julian has been lately and perhaps Miles should go spend time with him to "cheer him up". As Miles himself states: he's a lucky man.
The entire episode "The Begotten", where Odo plays Dad to a baby Changeling, especially the moment where it imitates his face. And bonus heartwarming in that the experience reconciles Odo with his own father figure in Doctor Mora Pol and it gives him back his shapeshifting abilities as a parting gift.
Virtually any moment with Odo and Kira after they get together is a heartwarmer, but if you want to know what real love means, watch this exchange from "Tacking into the Wind":
Garak: I am afraid he's been hiding the true extent of his illness from you for some time now.
Kira: I know.
Garak: You do?
Kira: I love him, Garak - did you really think I wouldn't notice?
Garak: Then why the pretense?
Kira: Because I also know he doesn't want me to find out about it. If he wants to put up a brave front and protect me from the truth, then fine. If that gives him one last shred of dignity to hold onto, then I'll go on ignoring what's happening to him until the very end.
Before all that, Garak notices the difficulty that Odo is having maintaining forms and is genuinely worried about him. When Odo cuts him off with a brusque, "If I don't want pity from the woman I love, what makes you think I want it from you?" Garak simply nods and walks away, leaving Odo with his dignity.
Any part of this story arc ("When It Rains", "Tacking Into the Wind", and "Extreme Measures) is chock-full of these, mostly due to the awesomeness that Kira and Odo bring to any scene. Special mentions go to the beginning scene of "Tacking Into the Wind" with Odo and Garak, the above-quoted scene, the end of this episode where Kira is so focused on Odo when he collapses that she seems to barely notice that she is being held at gunpoint, and the opening scene of "Extreme Measures".
Also, when Odo and Kira finally hook up. It's almost like a CMOF meets CMOA meets CMOH. Watch it here
Even more so, the fact that when Tain asked Garak if they were alone, Garak lied about the fact that Bashir was in the room, allowing him to stay and not only witness him reveal the biggest secret of his life but share in one of his most emotional and vulnerable moments. It's easy to miss this powerful unspoken statement of how much Garak trusts Bashir.
Garak and Bashir in "The Wire", when Garak asks Bashir to forgive him, which he does while holding his hand.
In "The Quickening" when Bashir seems to have utterly failed... but thanks to his antigen, a Teplan baby is born without the deadly blight, ensuring future generations will be safe.
In "A Call to Arms" Sisko's speech to the remaining people on the station promising to return before he beams over to the Defiant.
At the end of "Body Parts", Quark sits in the empty space that used to be his bar, all his possessions and business assets having been confiscated. Rom comes in to console him, and after a little bit... cue practically every non-Ferengi on the station coming up with contrived, mock-cynical, Quark-esque excuses to set up the bar exactly as it was.
The B-plot from the same episode, in which Kira allowed Bashir to transplant Keiko's unborn baby into her womb when Keiko is injured in a Runabout accident. The O'Briens welcome Kira into their extended family and invite her to stay with them for the duration of her pregnancy. Especially sweet when Molly calls her "Aunt Nerys".
From Dogs of War:
Quark Rom, I want to buy back the bar.
Rom: That's all right, brother, I'll give it to you.
Quark: I suppose you're going to let me keep the five thousand bars of latinum too.
Pretty much any time Quark shows affection for Rom counts. He may be a Jerk Ass, but family's family.
Quark risks his life to free his brother from the Dominion and has a key role in the success of the entire war mostly as a side effect of freeing his brother. He was willing to bribe the Dominion everything he had, he even got Zek to offer an official bribe from the Grand Nagus.
The end of "Soldiers of the Empire", in which Martok invites Worf to join his house.
The third season episode "Indiscretion" also has one of these for Dukat and his half-Bajoran daughter Ziyal. While he originally wanted to kill her to save his career, the episode ends with father and daughter embracing each other in a big, big hug. Especially heartwarming considering he actually gave up his career and social status by letting her live and openly acknowledging her as his daughter.
At the end of the Season 7 episode "Take Me Out To The Holosuite", Sisko's VulcanRival tries to rub in just how badly DS9 lost at baseball, even if they did manage to score a run. Sisko takes it in stride, enjoying the camaraderie and remembering how much he loved the game, then Sisko's team stand up for him, getting under the Vulcan's skin so bad it leaves him at a loss for words, showing how great the True Companions and Fire-Forged Friends tropes are.
Topping everything off with giving Sisko a new baseball signed by every player on his roster.
The baseball becomes a heartwarming Call Back in the series finale. After Kira assumes command of the station after Sisko defeats Dukat and the Pah-Wraiths but has to Ascend to a Higher Plane of Existence and go officially MIA as a result, she keeps the baseball on the desk: remembering the bonds forged then.
And on the greater scale of things, Nog lives up to that promise, becoming a model Starfleet officer who makes Sisko proud. In a way, even though Sisko was happy enough to let Jake become a writer and journalist rather that following in Sisko's footsteps as he'd hoped, Nog fulfilled Sisko's dream in Jake's place. Just to give it some perfect closure, one of Sisko's last acts in the final episode is to promote Nog to lieutenant junior grade.
Not to be missed is Rom's reaction to Nog wanting to join Starfleet. Quark forbids it and Rom, in a rare display of backbone, puts his foot down, tells Quark that he runs the bar, but Nog is HIS son, and wishes Nog good luck at Starfleet Academy.
Quark and Odo's final parting.
Quark: A-ha, I knew it! When I saw the two of you slip out, I said to myself, that no-good, misanthropic, cantankerous changeling is trying to sneak off the station without anyone noticing.
Odo: That was the idea.
Quark: Well, it's not gonna happen, is it?
Odo: Apparently not.
Quark: So - now that I'm here... is there something you want to say to me?
Odo: I mean, no. I have nothing I want to say to you.
Quark: You telling me that after all these years, after all we've been through, you're not even gonna say goodbye to me?
Odo: That's right. Nerys, I'll be on the runabout.
(Odo glares at Quark and gives a haughty, "HMPH.")
Kira: Don't take it hard, Quark.
Quark: Hard? What're you talking about? That man loves me! Couldn't you see? It was written all over his back. (He lifts his drink to toast Odo.)
Quark: It's like I said: the more things change, the more they stay the same.
Can we count the fact that Chief O'Brien goes through a yearly tradition of how he must suffer for an episode, but not once was Keiko divorcing him a plot? The O'Brien marriage goes through ups and downs, but they actually manage to spend the entire series Happily Married.
The wedding of Worf and Jadzia, and the tale of the first two Klingon, who instead of fighting each other, joined together and slew their gods, showing that nothing is stronger that two hearts together.
Special mention also goes to Martok talking some sense into Worf when he decides the wedding is off.
Martok: We are not accorded the luxury of choosing the women we fall in love with. Do you think Sirella is anything like the woman I thought that I'd marry? She is a prideful, arrogant, mercurial woman who shares my bed far too infrequently for my taste. And yet... I love her deeply. We Klingons often tout our prowess in battle, our desire for glory and honor above all else... but how hollow is the sound of victory without someone to share it with? Honor gives little comfort to a man alone in his home... and in his heart.
Jadzia and Lenarah Kahn in "Rejoined." Lovers in past lives, Dax and Kahn have been dealing with unresolved issue and flirting with taboo and after they kiss (itself a CMOH and Tear Jerker). While on the Defiant, Kahn is in the Engineering Bay and Dax is on the bridge when a plasma conduit is ruptured in Engineering. Jadzia wastes no time getting down there, and if Eddington says that if they don't vent the plasma into space, the entire ship will go. Dax walks over the plasma leak by putting a force field on top of it, pulls Kahn into the next room. Their exchange as they're holding each other:
Kahn: I don't want to lose you...not again...
Dax: Not again. Never again. Never again. Never...
While it may sound surprising, the episode "Take Me Out to the Holosuite" is probably one of the series biggest CMOH's. Watching Deep Space 9's senior staff pulling together all throughout the episode was very pleasant to watch. From everyone standing by Rom's side when Sisko kicked him off the team. Watching Quark of all people putting his heart and soul into the training, and the team. But the best part would have to be when Sisko gets Rom back in the game and Rom manages to score.
And both the "manufactured triumph!" bit and the team giving Sisko a baseball signed by all of them. Aww.
It's just a small thing, but still. Watching Koloth nursemaid Kor (despite his complaints that he would do no such thing) in "Blood Oath" was rather adorable in its own way. These are two battle hardened warriors who have watched each other's backs for almost a hundred years, so it kind of makes sense. Though it's also great on a meta level, since this is the first time the pair have ever been seen on screen together. The fans of the original series have been hoping for a team-up like this for years, and they finally got their wish.
The first season finale "In The Hands Of The Prophets" involves Vedek Winn stirring up conflict between the Bajoran and Federation crews, and when Sisko reaches the end of his rope, he starts speechifying.
Sisko: The Bajorans who have lived with us on the station, who have worked with us for months, who helped us move this station to protect the wormhole, who joined us to explore the Gamma Quadrant, who have begun to build the future of Bajor with us, these people know that we are neither the enemy nor the devil. We don't always agree. We have some damned good fights in fact, but we always come away from them with a little better understanding and appreciation of each other.
Dax meeting the Siskos in "Shadows and Symbols." Ezri is dealing with being an unprepared host for the symbiote and all the issues that go with the joining, Ben is trying to come to terms with Jadzia's death, his failure as the Emissary, and some Eldritch Abomination-provided hallucinations. But at the end of the day, they're still Sisko and Dax, and just being together again clearly helps both of them cope.
Kasidy Yates's return to the station. She's been in jail for about a year for smuggling food and medical supplies to the Maquis, and Sisko was the one who had to arrest her. When she comes back? It's like she never left and they never bring it up again. It's not a lack of continuity, it's that to Sisko, this is the woman he loves, and he doesn't care - she's done her time, and they can continue with their lives together.