Can my mother try and file for custody of my son?

It is difficult for a third party (grandparents, uncles, siblings, etc) to take a child away from his or her legal and biological parents in Georgia.  They must prove you unfit; not simply that you may or may not be a good parent, but unfit to the point that your child would be taken from you by DFACS for example.  It is a very high standard for your mother to meet.  Always underlying a custody case is the question, "what is within the child's best interest".  There is a presumption in the law that it is within the best interest for a child to be with his or her parents.  I would encourage you to seek local counsel.  There are many factors that could come in to play such as your current marital status and your age.