Goober was Gomer's cousin, mentioned as early as 1962 but not seen until season 4, episode 27 "Fun Girls." That episode was also the only time Gomer and Goober appeared on screen together during the series (though they did also appear together once on Gomer Pyle: USMC (1964) and in the reunion TV movie Return to Mayberry (1986)).
The newspaper on the wall of the sheriff's office has the headline "DEPUTY FIFE HERO IN CAVE RESCUE." This is a reference to Season 4, Episode 13, "Barney and the Cave Rescue," 14 episodes earlier in the season.
Goober imitates Cary Grant by saying "Judy, Judy, Judy." This line is commonly used by people who are imitating Cary. But Cary never used this line in any of his movies.
During the opening scene with Barney and Andy in the back room doing inventory. The crime chart made by Bob Rogers in the season 2, episode 2 "Barney's Replacement" can be seen on the wall behind them.
Barney Fife was called "Bernie" 16 times, total. Skippy said Bernie 14 times, Daphne and Sheriff Andy Taylor, once apiece. He was nicknamed Bern, once, by Skippy.