The locations of Beta Geminorum and Pollux on "The Explored Galaxy" wall chart
The Beta Geminorum system (or Pollux system) was a planetary system in the Mutara sector. This sector was a region of space in the Alpha Quadrant. The primary was named Beta Geminorum. The closest starbase was Starbase 12. Most of the planets, including Pollux V, were strangely devoid of indigenous intelligent life. However, Pollux IV proved to have life - Apollo. This was the homeworld of his species This species, the Greek gods and goddesses, was an ancient warp capable civilization that had visited Earth in antiquity. The location of Beta Geminorum in the Milky Way Galaxy was depicted on a Federation star chart in 2293. The Beta Geminorum system would, also, appear on a starchart Data and Picard were studying in Stellar Cartography in 2371. (TOS: "Who Mourns for Adonais?", Star Trek VI: The Undiscovered Country, Star Trek Generations)
Background
Michael Okuda, who created the chart The Explored Galaxy, believed that Beta Geminorum and Pollux were two distinct stars. So, he created two systems, the Beta Geminorum and the Pollux, and located them in the same region of space. Unlike the situation with another system, Canopus, where the canon never connected Canopus to Alpha Carinae, the canon did established that connection with Beta Geminorum and Pollux.
"The Explored Galaxy"
"The Explored Galaxy" was a map of charted space in the Alpha Quadrant. The Tholian Assembly and Romulus/Remus were identified in dialog as being located in this quadrant. (DS9: "The Search, Part II", "Call to Arms") This chart was also seen in several Star Trek: The Next Generation and Star Trek: Deep Space Nine episodes set in the 24th century, from the year 2364 to 2370. These were TNG: "Conspiracy", "The Measure Of A Man", "The Emissary", "The Mind's Eye", "The Game", and DS9: "In the Hands of the Prophets", "Cardassians"
Star Trek: Star Charts
Unlike Mr. Okuda, Geoffrey Mandel believed that Beta Geminorum and Pollux were the same system. This 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. In the mid-22nd century, this system was located in non-aligned space. In the late 24th century, this system was located in or near Federation space. (pgs. 60, 62)
Apocrypha
Pollux V was also featured in the video game Star Trek: 25th Anniversary.