In Godzilla (2014 Film) why is Godzilla hunting the MUTOs in the first place?
According to Wikipedia (which itself cites the book Godzilla: Art of Destruction):
David Callaham's first draft featured early versions of the MUTOs where they were established as ancient enemies of Godzilla but never established as to why. When director Edwards came aboard the project, he created a back story to bridge the gap between Godzilla's connection with the MUTOs. Edwards brainstormed the idea that, "When these Godzillas were on Earth, there was another creature that would kill them and lay its eggs inside their dead bodies. Therefore, if these creatures ever came back, part of their life cycle would be the ability to attract Godzillas to the surface to kill them for reproduction."
I believe that Godzilla hunted the mutos because of natural behavior, if we look at king of the monsters, Monster Zero was becoming the alpha of all the titans, so Godzilla set things right and killed Monster Zero, this made it so that he was the alpha. we see this again in Godzilla vs Kong, were Godzilla has to "kill" Mecha Godzilla because he was sending off signals saying "hey, I'm the new alpha now." leading to MechaGodzillas death, in Godzilla 2014, The MUTOs were threatening Godzilla's supremacy, so he killed them so that they would not do it again.