The trio correctly identified the square as the filming location of 'The Music Man' and 'The Gilmore Girls' (both already mentioned in this thread), and then Cathy says "and then didn't they film, um, Opie, uh, what's that, not Happy Days, um ...". She never finished the sentence and never mentioned Andy Griffiths, but it's fair to assume that that's what she was referring to as Ron Howard's character was Opie Taylor (and he was in 'Happy Days').
Exteriors for "The Andy Griffith Show" were mainly filmed at the RKO 40 Acres/Desilu backlot in Culver City, about 45 minutes south of the Warner Bros' backlot. I can't find any evidence of Andy Griffith filming in Burbank.
The video below shows what's now standing where the backlots were, and mentions several other shows and movies that were filmed there. It also shows the former Desilu Studios in Hollywood. There's a brief Dukes tie-in at around 12:32 when they show a still of James Best from the 1959 movie 'Verboten!'.