I am not familiar with the laws of Georgia; however in Texas, if no other court has continuing exclusive jurisdiction (meaning that your case hasn't already been heard in another court) you typically file a suit in the county where the child resides (see Texas Family Code 103.001). Georgia probably has similar laws and therefore, the appropriate venue (place where the father would need to file his suit) would be in the Georgia county where the child resides with you (assuming you are residing in Georgia when the father wants to initiate litigation).
Keep in mind that this is not an absolute rule. There are exceptions and everything is very fact specific. If the father does file suit against you, or if you decide to file suit on your own, I strongly advise you to contact an attorney in your jurisdiction to assist you. Best of luck.
This answer is intended for informational and educational purposes only and should not be considered legal advice nor constituting an attorney-client relationship. This attorney is licensed in Texas.
GA won't have jurisdiction until the baby has lived there 6 months, so if you move now, you do risk being hauled back to Texas for the case. Why not wait for six months then file for CS and custody in GA?
Better yet, file here and request temporary orders allowing you to move out of state. The judge probably will not grant that request, but you will get child support and a visitation order in place and maybe the baby's daddy will give in and let you go to GA by agreement.
This answer is intended for informational and educational purposes only and should not be considered legal advice nor forming the attorney client relationship. This attorney is licensed in Texas.