The ending of Son of Godzilla in which Godzilla lovingly embraces his son and keeps him warm as the island freezes over. It's the first time Godzilla shows any sort of compassion towards anyone.
Towards the end of Tokyo SOS, Kiryu (the original Godzilla's skeleton outfitted with mechanical parts) refuses the order to kill Godzilla. Rather than destroy his "son", he lovingly picks him up and flies him out to sea where they both rest, presumably until mankind breaks the keys off in the lock of science again. Though, he has time to leave one final message to one of the pilots-
In Rebirth of Mothra 3, the mortally wounded Rainbow Mothra is rescued by two prehistoric Mothra larvaes. Despite being from different ends of the evolution scale the two babies aid their descendant. Whether it was a burial or intended to give Mothra an upgrade the two babies' actions helped save the future. Also the end where all the parents are reunited with their missing children is actually pretty heart warming.
Say what you want about Minya, but when he steps between Godzilla and the humans at the end of Final Wars and manages to convince Godzilla to finally let go of his anger toward humanity... ::sniff::
Also from Final Wars, once Godzilla's beaten King Ceasar, Rodan, and Anguirus. He gathers the unconscious monsters into a pile and leaves. Now there are two moments that should be pointed out here. One, Godzilla could have killed the three where they fell but didn't. Two, Godzilla didn't just leave the three were they fell and instead brought gathered them all together.
It's not much, but Jet Jaguar shakes Godzilla's hand in absolute gratitude when he first saves him and then does it again once the battle is over and Godzilla prepares to leave.
Made better by the fact that Jet Jaguar's a frickin' awesome robot, and robots aren't particularly known for having emotions. Somewhere along the way, when he gained sentience, Jet Jaguar learned how to express gratitude.
This PSA made using clips of Godzilla and Minilla. Even Kaiju can be good fathers, too.
Despite Akane having put up personal walls around her, being the black sheep of the AMF, and being the only woman everyone sticks up for her after her teammate begins berating her in the open.
In an issue of Godzilla: Rulers of Earth, Godzilla is pitted against Destoyah, when Mothra shows up! For the most part, Big G and Mothra work pretty well together. When Mothra was kicked around, Godzilla always seemed to look angrier. When the fight was over, Mothra was hurt to the point where her sisters had to carry her. Before leaving, Godzilla roars to her, as if to say Thank You.
Similarly, in Ghidorah: The 3 Headed Monster, King Ghidorah shows up and kicks Mothra's ass. Godzilla sees it and goes absolutely nuts. Shows you how much he cares about his friends. Who would've thought a radiation-filled angry invincible dinosaur would be such a Nice Guy?
A bit of a darkly example, but in the 2014 Godzilla film, the reunion on both Male and Female MUTOs after being apart for millions of years is certainly touching.
Through out the new film there are scenes of family caring for one another and for strangers, Ford Brody for his wife and son, Joe Brody for his own wife and Ford, the Japanese tourists for their son and Ford keeping an eye on him. Then the Muto have their own for their own lost children once Ford, understandably, destroys their nest and the Muto are both distressed over the loss of their eggs.
In Godzilla vs. Biollante Asuka goes to comfort Kazuhito after Agent SSS 9 is killed by giving him a hankerchief and complimenting him.
"You were like Batman." (Japanese version.)
"Great job, Superman.''" (U.S. dub.)
In the middle of the climatic battle of GMK Yuri makes a plee for people to support the last standing monster, King Ghidorah, against the vicious Godzilla and for all the soldiers that had and were ready to give their lives. Important since the reception to the Guardian Monsters were a bit mixed and the message of the film is to never forget your veterans, or they possess the corpse of a super dinosaur mutation but still don't forget your veterans.
A meta example: the reason we are getting a new Toho-produced Godzilla film is because of the box-office success of the 2014 film and the revival of the fanbase.