Whenever Randy cheers on the "Big Guy" during a fight. While everybody but Nick seems to see Godzilla as a frightening monster, Randy thinks he's awesome and is almost as protective of the Big Guy as his adopted parent. Later on, you can occasionally see the rest of the team cheering or showing relief whenever Junior shows up.
Despite the occasional clashing between Nick and Audrey over what's important at the moment (science and journalism, respectively), the scenes where they show concern for and reconcile with each other is equally heartwarming, especially for couples that are/have been in similar positions.
S1E01 "New Family, Part 1": When the team (sans Monique, who hasn't joined yet) flee in terror from a juvenile Godzilla and hide inside their headquarters, Godzilla reacts as a faithful dog would: he scratches at the door, turns around a couple of times, and lays down with a sad whine. Daaawww...
Also, we see that Godzilla got his own statue for his Heroic Sacrifice by protecting millions of people from the DRAGMAS, though it's also a Tear Jerker.
S2E02 "Cash of the Titans": Once the team confirms that Godzilla's alive and in one of Spiel's holding pens, this line from Nick.
Nick: Godzilla's always saving us. Time to return the favor.
S2E04 "Protector": Despite being temporarily incapacitated by Norzzug's blinding sandstorm attack, Godzilla sticks a claw out just in time to shield the team from being crushed by a scaffolding tower that got knocked loose by Norzzug's wing.
S2E05 "Freak Show": A subtle one. Hicks, of all people, asks if Godzilla is dead in a worried tone after Medusa severely dehydrates the Big Guy. When Hicks sees that Godzilla's too weak to move, he orders his tanks to push Godzilla into the reservoir so he can drink.
What makes this truly heartwarming is that Hicks doesn't have to be asked; while H.E.A.T. are trying to think of what to do, Hicks just pulls out his radio and starts giving orders to help the big guy out.
S2E16 "Ring of Fire": Both father and son, one right after each other. Nick's determined look and resolute "I'll protect Godzilla" shows the level of his devotion to his adopted son. When Nick starts faking his distress, Godzilla was fighting the Fire Monster and barely hesitates before swatting the Fire Monster aside with his tail to rush to Nick the moment he heard his adopted father's voice. When the eventual nitroglycerin explosion dislodges Nick's safety line, really putting him in danger, Godzilla catches him with the top of his head. The following line pretty much seals it on showing how deep their parent-child bond goes.
Nick: Knew I could count on ya.
As Mendel struggles to guide a nitroglycerin-ladened and wrecked N.I.G.E.L. towards the Fire Monster all the while lamenting his decision and plan, Randy comes help despite the danger of also being blown up.
Randy My point is, we're partners! And partners split everything, 50-50. Glory and responsibility.
S2E18 "The Ballad of Gens Du Marais": After Godzilla yanks back the Swamp Monster from the fish processing plant, he pokes his head through the roof to check on Nick, as if to say "Hey, Dad! You're okay!" Nick has a relieved smile on his face as well.