When mailing my deceased mothers tax return for the state of GA do I need to complete a form similiar to the 1310 federal form?

Yes, you do. 

You need to use Form GA-5347, Georgia Department of RevenueDeceased Taxpayer Refund Check Claim Form to  claim  a  refund  on  behalf  of  your  deceased mother.

The wording on the form is kind of funny, but this is what you'll need to mail in with your mother's Georgia return.