It depends. Gun rights are generally only taken away from persons who become convicted felons. If your conviction was for felony possession of marijuana and you did not use First Offender or Conditional Discharge type of pleas then your right to purchase a gun would be restricted. You also cannot purchase a gun while you are still on First Offender probation.
If, however, your conviction was for misdemeanor marijuana, this should not restrict your right to purchase a gun. You may be restricted from getting a concealed weapons permit but not actually buying a weapon.
If your charge was a felony and you wish to get your right to buy a gun back, you can apply for a restoration of your rights with the Georgia Board of Pardons and Paroles. An attorney like myself who is an expert in criminal defense and appeals can help you figure out what your rights are and whether you have the ability to lift any ban on your right to bear arms.
You should consult with an attorney about your specific case.
Attorney at Law