At what age in Georgia can a child decide about not visiting with the non custodial parent?

If there is a court order granting the father visitation, you have to do your part in "making" your son go. There is not an age in Georgia when a child can decide not to visit. As a practical matter, it may become almost impossible to force an older child to visit (i.e. a 16 or 17 year old) but if your 12 year old stops visiting, you could well be held in contempt.

At his age he can’t decide to not go to the visits. If he does you may find yourself in court and in contempt.