As part of the Galactic Cultural Exchange Project, the Karidian Company of Players performed an Arcturian version of Macbeth on Planet Q in 2266. At that time, Doctor Thomas Leighton, a resident of Planet Q and old friend of Captain James Kirk, sent a signal to the USS Enterprise, informing Kirk that he had discovered an extraordinary new synthetic food, which would end the threat of famine on nearby Cygnia Minor. It was soon discovered, however, that Leighton had lied about the discovery in order to lure the Enterprise to Planet Q to convince Kirk that he had found Kodos. (TOS: "The Conscience of the King")
In 2328, this was a point of embarkation for passengers traveling to Tarsas IV via commercial transport. Lolita Fatjo traveled there on the SS Manoa, Jimmy Berndt traveled to Planet Q on the SS Kogin, and John P. Farrell and Carl Banks traveled there on the SS Wisconsin. (TNG-R: "Inheritance", okudagram)
A city on the planet's surface was scripted to be "a strange, modernistic city," and the script then went on to remark (in a bracketed note), "The CUT OUT from PILOT ONE, if possible." Indeed, the Planet Q city was ultimately represented with a futuristic skyline that had originally been used to depict Mojave in the first Star Trek pilot, "The Cage". (The Art of Star Trek, p. 22)
According to Star Trek: Star Charts ("United Federation of Planets I", "United Federation of Planets III") and Stellar Cartography: The Starfleet Reference Library ("Federation Historical Highlights, 2161-2385"), Planet Q orbited a F-class star. In 2378, this planet was listed as a Federation member.