Can my fiancé sign my sons birth certificate at birth if he is not the biological father,the biological father left when I was only 2 months pregnant. Is it illegal for me to put my fiancé on his birth certificate if he's not the biological father?

And is it illegal for him to do so in the state of Georgia

It is illegal to falsify a birth certificate. Having said that if you were married at the time of birth then he would be the presumed father and that makes it legal to do so.

The presumed father need not be the bio father but you must be married before the birth to be granted that status.

The parents don't actually sign the birth certificate in any event.

It is illegal to falsify a birth certificate. By naming your fiancée as the father, you would be committing perjury. This would also be true if your fiancée signed an acknowledgment of paternity.

You can marry before the birth, maybe even within a year after the birth and have him become the presumed father.