Shortly before Jim Nabors left The Andy Griffith Show to do his own show, there was an episode in which Gomer and Goober walk into the Court House, where Gomer introduces his cousin Goober to Andy. Gomer then prods Goober into doing his famous Cary Grant impersonation for Andy - "Judy, Judy, Judy!" (Of course, he didn't sound anything like Cary Grant.) I believe this is the only episode in which Gomer and Goober appeared together.
They appeared together, in one episode, in the Gomer Pyle, USMC TV show.
Funny how we would remember that. :-)