You were convicted of a non-gun related no violent felony and have completed probation and paid your fines Can you own or possess a firearm in Arizona?

The Federal law has an absolute prohibition of felons owning or even handling firearms
According to Arizona law, you must have your felony conviction expunged, or set aside, in the jurisdiction you were convicted in. Once this is completed, you must petition to the court, a minimum of two years after the fines are paid, to restore your gun rights.

If you possess a firearm without doing this, you will be charged with a felony.