Ickis and Krumm comforting the heartbroken Oblina at the end of "Smile and Say Oblina". Always makes me go "Awwww..."
Ickis bonding with a human baby in one episode.
Ickis' bonding with Fungus in "Puppy Ciao"; how devastated he gets when he has to send Fungus back to his human owners is somewhere between this and Tear Jerker territory.
The Gromble gets one where he encourages Ickis that he didn't fail in his quest for the Holey Pail, having been sent on that same quest himself 400 years ago, and then lets him sleep through class.
The episode "Monster Blues" is pretty heartwarming. Ickis befriends a blind blues man named Mr. Robinson living in an old condemned building. There's something about whenever Ickis makes friends with a human. In this case he and his new friend vent their woes by singing the blues. Ickis even helps the old man save his building from getting torn down and saves his life. The old man tells Ickis he just has to face his troubles and that he's a scary monster after all. After everything he gives Ickis his harmonica and later we see the building is still there. They just decided to build the office building in top of it.
In "A Friend Indeed" Ickis bonding with a monster named Plastoog who was ostracised by his peers from the Academy many years ago because he had a human face, he only had one friend at the Academy and that was The Gromble
The Gromble and Plastoog reuniting as friends after so many years
In "Super Ickis" The Gromble dresses up as Ultra Monster to save Ickis and Krumm when they get in over their heads and were almost squashed by a human, and gives them an inspirational speech about staying in school and how scaring comes from the heart, not from a costume. To see him do that for two of his students shows what lengths he'll go to help them in their time of need is really sweet and shows his 'well meaning, fatherly' side to his students whether they know it or not.
Slickis telling Ickis that he has the makings of a truly monstrous monster at the end of "Chip Off The Old Beast." The next episode of the season, "Where Have All The Monsters Gone" actually has Ickis save not just his class but possibly the entire monster world.
The end of the ep "Blind Love, Monster Love"
Ickis: Nobody's perfect but you were pretty close.
The end of the ep "Spontaneously Combustible"
Even before that, the Gromble shows his fatherly side not once, but twice during that episode. The first time is when he uses a history lesson about famous monsters (all of whom were afflicted with the same condition that Ickis has) to point out that these famous figures became an important part of the monster world despite their condition (and, by extension, implying that Ickis could do the same). The second time comes when he chews out everyone for believing the rumor mill and ostracizing Ickis:
Gromble: ...I told you that [Ickis] wasn't a danger to any of you! But you listened to the rumors and were afraid. Do you know what else is afraid of something they don't understand?!