@Eliharrison56,
I believe that in Georgia, a child has to be 14 to be able to decide which parent to reside with.
I would imagine as there is no Child Support in place if her Father is happy to pay her way and have her move in with him it wouldn't matter.
However, I do believe that Child Support ends at 18, so I would imagine if her Mother is upset over this move, she would be able to apply for Child Support but not necessarily stop her from moving, therefore the support I would imagine wouldn't go through as she no longer lives with her. She could request back Child Support perhaps.
Are you worried that her Mother will stand in the way? It seems that her Mother has cared for her all this time, I hope they have a good Bond together as obviously there would have been some tough roads for her, to have done this without your girlfriend's Fathers support.