Quirk in the law: the 1986 federal amendments provided that a State restoration of rights after a felony convicted acted to restore Federal rights to own a gun, unless the State law expressly provided otherwise. (I know -- I wrote that clause while working in Washington).
Arizona's rewrite of the criminal code in 1977 simply provided that, after a first conviction, civil rights were restored automatically at the end of probation. In 1988 the law was amended to provide that this did not apply to firearms rights -- you had to file a motion to get them restored. So if a conviction on a first offense was entered prior to 1988, rights were automatically restored and you probably don't need a motion. (The bill was signed on June 21, 1988 -- I haven't checked the effective date, which would have been after the legislative session ended that year). You might want a restoration anyway (1) to be sure and (2) I'm not sure that the background check computers know this detail of Arizona legal history.