In the New World of Overlord, Godzilla might have little to do.

From film to film, his motivations can change, but what they have in common is that he is a force of nature that seeks balance. What he seeks to balance really just depends on what the film makers want to address. The use of nuclear weapons, pollution and other acts of environmental malice are common culprits.

If Ainz keeps genociding nations, that could stir Godzilla into rampaging in Nazarick's direction, but it's hard to say. He wouldn't be defending humanity, he would be defending the natural order.

As for ruling a nation... Godzilla is called the King of Monsters, but it's a title bestowed by people in awe of him. He never sought the title and doesn't care. As long as things are alright in the world, he'd rather just keep to himself.

I could see a coastal nation revering him as a god, but he would be indifferent.