The Talos star group.
The Talos star group was a star cluster named after the Talosian planetary system. This star group was located in the third quadrant of the vernal galaxy. This star group, located between Rigel VII and the Vega colony, was in the Alpha Quadrant.
By the mid-23rd century, this cluster was unexplored and there were no Federation starships or colonies here. The only expedition into this region, a survey by the SS Columbia, had culminated in the Columbia disappearing under mysterious circumstances in 2236. Eighteen years later, in 2254, the USS Enterprise, under the command of Captain Christopher Pike, on a search-and-recovery mission for survivors of the downed survey ship became the second ship to enter this star group. (TOS: "The Cage", "The Menagerie, Part I")
Within thirty years of this mission, the Federation had expanded into this star group, and had established at least one colony in the Talos system. By the late-23rd century, the Talos star group was firmly within the Federation fold. (Star Trek II: The Wrath of Khan; Star Trek VI: The Undiscovered Country)
Background
The image of the Talos star group was a color astrophotographic exposure of M45, the Pleiades Cluster. The exposure was relatively short, and so was lacking detail of the characteristic Merope Nebula. However, the nebula does give a blue halo to the stars visible within the photograph.
The vernal galaxy information comes from the "For eyes of Starfleet Command only" file read by Kirk in "The Menagerie, Part I". It was unknown what vernal galaxy was, exactly, and how its quadrants are arranged. This reference system was never used again. Dialog in "The Cage" clearly states Earth and Talos to be in the same galaxy. According to Pike, the Enterprise originated from a stellar group at the other end of the galaxy relative to Talos.
According to the Star Trek: Star Charts (pgs. 34, "United Federation of Planets III"), the Talos Star Group was located in or near Federation space, in the Alpha Quadrant.