"I dare to dream that she might even care for me..."
When Esmeralda frees Quasimodo after he being 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.
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).
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 Esme 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 Esme 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"
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.
Right after Frollo's fall into the lava, Quasimodo seems to follow the same fate when Esmeralda looses the grip on him, but luckly Phoebus grabs him in time and saves him. How does Quasi thank him? With a smile, and then an hug. Cue Phoebus patting Quasi's back, and smiling as well.
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.
Quasimodo's mother; she literally died trying to save her son.