Beta Geminorum (also called Pollux) was the primary of the Beta Geminorum system and located in the Alpha Quadrant. Its location in the Milky Way Galaxy was depicted on a Federation star chart in 2293. (TOS: "Who Mourns for Adonais?"; Star Trek VI: The Undiscovered Country, production art)
In The Explored Galaxy, Beta Geminorum and Pollux were depicted as two distinct stars, even though they were established to be synonymous in "Who Mourns for Adonais?". In the star chart, both Pollux and Beta Geminorum were located between the Tholian Assembly and Romulus. Both were identified in DS9: "Call to Arms" as being located in the Alpha Quadrant.
According to Star Trek: Star Charts (pg. 60), the Beta Geminorum system was located in the Beta Quadrant. Beta Geminorum was a K-class star with a magnitude of +3, which was ten times brighter than Sol.