There are two known references to the Beatles on Family Guy.
In the epidode Stewie Kills Lois, Brian gives Lois tickets for a cruise to go with her, but she believes it is to go with Peter. Brian exclaims, "Great, has been more useless than the songs that composed Ringo". A scene follows with Paul McCartney , John Lennon and George Harrison Harrison in the study. Ringo arrives announcing that he has composed a song, others will applaud, and then Paul says he is going to put it on the fridge to view it every day.
In the first Star Wars Family Guy special, Lois (as Princess Lea) becomes angry at Peter (Han Solo) for driving into an asteroid field. Peter/Solo exclaims, "It was either this or the Strawberry Fields!" The scene cuts to Yellow Submarine-style John, Paul, George, and Ringo in space, surrounded by giant strawberry's. Paul states "You know what I'm really getting into, is one-legged chicks." John replies, "Yeah, what's your new girlfriends name Paul, Peg?" Paul becomes annoyed, and Ringo adds in "How about Eileen?" John replies "HA! Ringo got it!"