Would there be any problems on moving out if she turns 18 in papers this year but her parents tell her she’s turning 17?

"Her papers" and "what her parents tell her" are phrases that indicate to me someone wants to play games. When one is 18 they can move anywhere they want, and can no longer expect any support from their parents if they do so. Your gf knows how old she is. If she doesn't, you should not have a relationship with her until she confirms her age and gets a checkup to make sure she is a mentally competent adult.