I have a question about a Godzilla fanfiction I want to write

If you want to specifiy in the exact Godzilla incarnation in the synopsis to help readers picture the action in their head, that's fine. I don't think it's important however, since more readers would probably rather fill the image with their favorite appearance. Within the story itself, I wouldn't name drop films to describe characters or objects. It would be more effective to use original descriptive adjectives like you would find in an traditional storytelling.