Can a convicted felon in Georgia own a BB/Pellet gun. I am off probation and fines have been paid,

It depends. If you pled First Offender, your probation is over, and you've been formally discharged under the First Offender statute, then yes.

However, if you're a convicted felon then per Graham v. State
236 Ga.App. 673 (1999), my legal opinion is "No." In that case, the Court of Appeals found that the evidence was sufficient to convict the defendant for possession of a firearm by a convicted felon while possessing a pellet gun.