Legal Question in Civil Litigation in Georgia

yes

The agreement was between your mother and the lady. However, you actually provided her a baby-sitting service and performed your part of the bargain so you can recover in quasi-contract to the extent no actual contract exists. .

Since the lady has not paid you, I would not baby sit anymore for her. And I would send her a final letter asking her to pay up or else you will bring legal action in ten days or something like that. The reason is that some judges like to see some written evidence that you have tried to resolve the matter before coming to court. The letter does that. Send it via certified mail so she cannot deny that she received it. If you have already done this and she does not pay, go to the magistrate and fill out a civil complaint form for money owed.