The Omega system was a planetary star system. This system included Holberg 917G, a small planet with two moons, purchased by Mr. Brack in 2239. (TOS: "Requiem for Methuselah") The system was located in the Alpha Quadrant. Its location in the Milky Way Galaxy was depicted on a Federation star chart in 2293. (Star Trek VI: The Undiscovered Country)
The USS Enterprise visited the system in 2269, on Stardate 5843.7, in order to obtain the ryetalyn from Holberg 917G that would combat the deadly Rigelian fever threatening its crew. (TOS: "Requiem for Methuselah")
Holberg 917G, a planet in the Omega system
The location of Holberg 917G, a planet in the Omega system, and Omega IV on "The Explored Galaxy" wall chart
"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 I", "Call to Arms") The 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".
Based on the "The Explored Galaxy" star chart, there were two systems that shared the name Omega. The first system contained the planet Omega IV and the second, which was closer to Sol, contained the planet Holberg 917G.
According to Star Trek Maps (page 28) though, both planets were in the same Omega system.