In The Swimming Pool, Fred ends up getting thrown in jail on the night of his birthday due to a botched misunderstanding (specifically, mixing up a real cop with one of his pals hired to scare off Barney from the pool), but Barney shows up at the window and gives Fred a piece of his own cake that he never got to eat. Fred is literally in tears from this token of kindness.
On top of that, Barney also bails him out of jail the next day (Wilma gave him the money, but it's the thought that counts).
It's small and short, but Bill and Joe's cameo at Pebbles and Bamm-Bamm's wedding in I Yabba-Dadda-Do.
The song that plays during the Flintstones and Rubbles taking care of the twins in Hollyrock-A-Bye-Baby.
The flashback scene to when Pebbles and Bamm-Bamm were introduced to each other.
Seeing the normally loud, loutish Fred trying his best to keep Wilma rested during her pregnancy is very sweet. This reaches its climax in "The Dress Rehearsal," when Wilma gives birth to Pebbles. The excitement and joy on Fred's face are adorable, especially when he rushes to the maternity ward window and mistakenly believes that all of the babies inside are his.
Ending of Dino Goes Hollyrock, where Fred realizes his mistake, burns the money he received and puts out the candle with his own tear thinking he'll never see Dino again. Then Dino comes back and it's all the more reason for tissues.
During the lead up to Pebble's birth—when Fred and Wilma didn't know if their baby was going to be a boy or a girl—Fred buys a bunch of baseball equipment for the baby, on the hopes that "he" would be a potential baseball star. When Wilma points out that "Junior" (as Fred called the baby) might end up being a girl, Fred thinks it over for a second, and then says that it'd be great—she'd be the first female player to make the big leagues. It shows that Fred is just happy with the idea of being a father, and will be proud of his child no matter what.
One episode following Pebbles' birth consists of Fred and Barney babysitting duties conflicting with a boxing match, and the two gradually convincing themselves that such an event could make a good bonding moment for her and Fred (of course without Wilma finding out). The catcher however is, unlike so many other antics, this doesn't backfire. Pebbles genuinely wants to watch the match with her dad and her presence only sets off an adorable Cuteness Proximity with the competitors.
The Flinestones On the Rocks see's Fred and Wilma having marital problems that are so serious, with each forgetting about their own anniversary, that Barney and Betty take them on a vacation so that they can work things out. During a good portion of the trip Fred and Wilma are at each other's throats and Fred is especially mean to Barney when, as Barney points out, his friend is paying for the entire trip. However, around halfway through the film there's a segment where Fred takes the time to express his sincere gratitude to Barney for providing him and Wilma with the opportunity. He acknowledges that he can be a loudmouth jerk and act as if he doesn't like Barney, but (with tears in his eyes) admits that he does consider Barney to be a friend and even says that he loves him. For a character like Fred Flinstone this is probably one of the most heartwarming moments in his history. Though it's slightly ruined due to the fact that Barney never actually hears it.
Fred giving Stoney the honor of putting the star on top of the tree in A Flintstone Family Christmas, now accepting him as part of the family.
Between his tiptoeing and how terrified of his wife's wrath he is, Fred giving his family savings to the Rubbles so they could adopt a child was really heartwarming. It's also what cools off Wilma, as Fred usually isn't capable of something so kind (or that complex for that matter).
I was always the richest man in the world; I just never knew it.
In The Flintstones in Viva Rock Vegas Wilma brings her friends to her father Colonel Slaghoople's birthday party. It's the first time they discover that she's an extremely wealthy heiress, and as such, the party is fraught with tension. When Wilma's alone for a while, her father enters with a box:
Colonel Slaghoople: I brought you a little something for your birthday!
Wilma: Daddy, it's your birthday.
Colonel Slaghoople: Oh! Well, these won't fit me...
(He opens the box to reveal a pearl necklace—Wilma's most iconic piece of jewelry.)
Wilma: Oh, Daddy...
Colonel Slaghoople: I know you don't like showy things, but...Wilma, dear, I want you to know that no matter what you do, your daddy will always love you, and couldn't be prouder of you.