The Beta Geminorum system was a planetary system in the Alpha Quadrant. Its primary Beta Geminorum was orbited by at least five planets. The closest starbase was Starbase 12. (TOS: "Who Mourns for Adonais?"; DIS: "Magic to Make the Sanest Man Go Mad")
In 2257, during the first Federation-Klingon War, this system was occupied by the Klingons. (DIS: "The War Without, The War Within")
When surveying Pollux V and the rest of the Beta Geminorum system in 2267, the USS Enterprise detected a statistically unexpected absence of intelligent life at first. However, the ships's crew soon encountered Apollo, the last member of his species of ancient intelligent humanoids, on Pollux IV. (TOS: "Who Mourns for Adonais?")
In 2293, the location of Beta Geminorum in the Milky Way Galaxy was labeled in a star chart that was in Captain James T. Kirk's quarters aboard the USS Enterprise-A. (Star Trek VI: The Undiscovered Country, okudagram)
In 2371, the location of Beta Geminorum was labeled on a star chart Data and Picard were studying in stellar cartography on the USS Enterprise-D. (Star Trek Generations, okudagram)