"I dare to dream that she might even care for me..."
When Esmeralda frees Quasimodo after he's been tied down and humiliated by the crowd. Also counts as a Crowning Moment of Awesome, as she was defying Frollo by doing it AND (for that time) got away with it.
Just before that, when some of the people are attacking Quasi. Phoebus looks at the attackers, disgusted, and requests permission from Frollo to step in and put a stop to it.
Towards the end, when Quasimodo walks out of the cathedral, expecting to be ridiculed and shunned like he has been all his life. Then a little girl approaches him, touches his face, and gives him a hug. Say it with me: D'awww.
The moment before that is very sweet, too; Esmeralda wordlessly beckoning Quasi out into the public, all with the warmest smile that seems to say that things will be alright to Quasi (and from the layout placing, the audience).
To make it even better, there's some foreshadowing related to this scene. There's a scene in the movie earlier where Quasidmodo is talking to his gargoyle friends, angrily asking "What am I supposed to do? Go out there, save the girl from the jaws of death and have the town praise me like I'm some sort of hero?!" Everything said in that sentence comes true in the final act of the movie.
Plus add in this little fact as well: After rewatching it, the little girl? She was at the beginning of the movie! She knows about Quasimodo's mother's death from Frollo and is likely seeing him for the first time, afraid of them from the Feast of Fools incident. So she was the one who helped Quasimodo get over his fear and show him that they're not so bad. Even adding that she and Clopin watch him go off together, with him playing around with one of the hand puppets from the beginning of the film from when he told Quasimodo's story.
Quasi blessing Esmeralda and Phoebus' relationship. Despite most saying that this breaks the movie's "looks aren't everything" message, this shows a lot of growth in his character in being able to let Esmeralda love someone else and make peace with Phoebus. Extra bonus for the melody from "Heaven's Light" playing.
Arguably it is not broken as the characters are loved for their personalities. Phoebus is shown to have inner beauty himself and slight character development from a reluctant pawn of Frollo to someone who disobeys orders to rescue an innocent family. He is attracted to Esmeralda at first sight but loves her for spirit and energy, and she loves him because he is kind at heart. To add to the Heartwarming list, Phoebus completely credits Quasimodo for finding the Court of Miracles when he notices how Quasi looks forlorn at being ignored.
It's even more heartwarming when you realize it does, in fact, keep with the movie's real Aesop: the comparison between Quasimodo and Frollo. By blessing Esmeralda's relationship with Phoebus, Quasi proved that he's the real man despite looking like the monster. He's happy knowing Esmeralda is happy.
"God Help The Outcasts"
Part of the song has the various parishioners asking for various blessings, such as wealth, fame and love to name a few. Esmeralda, who has many, many reasons to ask God for help for herself at that moment and would be justified in doing so...instead prays that God help those who are far worse off than she is, saying she can take care of herself. It's a subtle moment but a very powerful heartwarming moment.
When the little girl brings Quasi into the crowd, and Clopin calls "Three cheers for Quasimodo!" And he's carried out into the crowd. He's happy, he has his freedom as he desperately wanted, he didn't need a girlfriend to get his happy ending.
Quasi's interaction with the baby bird at the beginning after the prologue. According to the commentary, the makers wanted a more humane Quasimodo, as opposed to the monstrous ones of past adaptions. They got it. When the gargoyles are trying to convince Quasi to take part in the Feast of Fools instead of watching, they even use his earlier "No one wants to be cooped up here forever" line he used to encourage the bird.
Knowing the story a bit can help with the bird scene, as his line "No one wants to be cooped up here forever" clearly refers to himself moreso than the bird.
Esmeralda, who is safe behind the parapets, still desperately clinging onto Quasimodo, to prevent him from falling, even though Frollo stands above the both of them sword in hand, ready to strike both down. In this instant she is willing to die alongside her friend, even though she could easily save herself. The very definition of friendship.
So while some people complain that Quasi Did Not Get the Girl, in a way, with her devotion to her friend placing his life above her own, which come on if that isn't done out of a form of love then what is, he kind of did except in a way it's even better, Quasi got a life-long friend in the end.
Right after Frollo's fall into the flames and chaos, it seems that Quasimodo will follow the same fate when Esmeralda looses her grip on him, but luckily Phoebus grabs him in time and saves him. How does Quasi thank him? With a smile, and then a hug. Cue Phoebus patting Quasi's back, and smiling as well.
Quasimodo's mother; she literally gave her life in trying to save her son (it's nice to know that Quasi's biological parents, especially his mom, still loved him despite his appearance).
The scene with the Archdeacon stopping Frollo from drowning Quasimodo. First off, the way he handled the situation by telling him that God had just seen him murder an innocent woman and will pay the price with his eternal soul is a Crowning Moment of Awesome. Second, it shows what a compassionate man he is in contrast to Frollo, caring for a stranger and her infant. Plus, it looked like he was going to bury her body instead of leaving it on the steps.
The way Clopin defuses the situation after Quasimodo is identified by a villager at the Feast of Fools is heartwarming in its own way.
He was also going around trying to get Quasimodo to be more involved when he first catches him. While this could be seen as mean, he is at least trying to get someone who seems to be a wall flower of sorts to get involved and adds him to all the chaos. It's nice that the King of Gypsies wanted everyone to enjoy the feast of fools. Plus during the second part of the Topsy Turvy, he immediately starts to build up Quasimodo's confidence. If not for the soldiers, everyone would have shown Quasimodo a lot of kindness because, despite how he looks, he is still the King of Fools; so that'd make him a local celebrity!
Even earlier during the Topsy Turvy song, Quasimodo accidentally tumbles into Esmeralda's tent. Though at first she sounds annoyed, due to having accidentally torn down the curtain that had been covering her, she quickly becomes concerned if he's OK since he fell to the floor. He simply attempts to leave the room, but she insists on checking if he's OK, eventually seeing his face (which she assumes is a mask), and only reacting with relief that he isn't hurt before warning Quasimodo to be more careful.
The scene where Phoebus first talks to Quasimodo right after helping Esmeralda escape. Quasimodo is already pissed that a soldier is in his bell tower and tries to be intimidating in order to drive him out of the church demanding him to leave, waving a torch wildly at him. Phoebus, though scared manages to explain why he wanted to talk to her which causes Quasimodo to stop.
Phoebus: Tell her I didn't mean to trap her here, but it was the only way I could save her life. Quasimodo: I'll tell her if you go. Phoebus: Oh, and one more thing: tell Esmeralda - she's very lucky. Quasimodo: Why? Phoebus: To have a friend like you.
The deleted number, "In a Place of Miracles", is packed with heartwarming moments. After Esmeralda saves Quasi and Phoebus from the gallows in the Court of Miracles, she and Phoebus confess their love for one another as they begin to dance together. At first, Quasi is visibly heartbroken, but eventually, Clopin invites him to celebrate with the Gypsies, leading Phoebus to give him a toast as thanks for bringing him and Esmeralda together. Touched, Quasi finally accepts their love and makes a toast "to love, and friendship." Clopin's touching comment afterwards is the icing on the cake.
Clopin: Spoken like a true Gypsy.
Phoebus'sintroduction. Esmerelda has earned some coin through her dancing, but some thuggish guards think she is a thief. Phoebus knows she's innocent (after all, he paid for her dance). After she escapes and runs, Phoebus blocks their way with his horse so she can escape. When he notices Esmerelda dropped some of her earnings in her flight, he picks them up and gives them to a beggar woman (who happens to be Esmerelda in disguise).
The Disney sequel:
For all of its flaws the sequel does have its own genuinely heartwarming moments, like Quasimodo's relationship with Zephyr and Madellaine admitting her love for him.
Bonus points for the Meaningful Echo Madellaine uses to convince Quasimodo that she's on his side and wants to help: "There's more to me...", a phrase that's as true about him as it is about her.
The Musical Production (Original German/2015 American Production):
Sort of a meta example, but Stephen Schwartz and Alan Menken (the songwriters of the movie) created a song that had to be cut from the movie. The song was a love song between Phoebus and Esmeralda and also gave focus to Quasi coming to terms with his ugliness and his slim chances on having Esmeralda. It was a sweet song that unfortunately had to be cut from the film. They were personally heartbroken that it was cut. But now the song has been reworked in the Broadway musical version. The lyrics have been reworked to include all of the gypsies and focuses more on the tragedy that Quasi will be alone forever, but in the end, it's real sweet that such a beautiful song has been able to end up in the musical after all.
That song is, of course,"In a Place of Miracles" .
During "Top of the World" Esmeralda looks down at Paris from Notre Dame's highest tower, but quickly steps back from the edge of the balcony because of her fear of heights. Quasimodo then proceeds to show her tricks and to balance on the balustrades, even joking a bit by pretending to stumble. At first, Esmeralda is scared for him, but he manages to put her at ease to the point that she even sits on the balustrade while stretching out her arms. It shows how caring Quasimodo is and that he is actually pretty in tune with other peoples emotions despite having almost never interacted with anyone besides Frollo.
Combined with a Tear Jerker, from the "Top of the World" reprise:
The finale "Olim" reprise is sung by a solo chorus soprano, explicitly referred to as Florika (aka Quasimodo's mother). The implication being that even a so-called "heathen" can make it into heaven through Heroic Sacrifice and the power of love. Take That! Frollo!