Gamma Canaris region
The homeworld of the Companion was a planetoid in an asteroid belt located in the remote Gamma Canaris region. This region was located in the Alpha Quadrant. (TOS: "Metamorphosis"; Star Trek VI: The Undiscovered Country, production art)
According to Spock, the planetoid was possibly a remnant of a planet breakup. It was one of 7,000 bodies of sizes running from types A to N in the asteroid belt, 34 percent of which had atmospheres of types H to M. The composition of the planetoid was iron and nickel, and it had an atmosphere that consisted primarily of oxygen and nitrogen with traces of krypton, argon, and neon. The general surface temperature was 75 °F. Gravity was 1 Earth standard. (TOS: "Metamorphosis")
Some time after 2119, a non-corporeal lifeform native to the planetoid, the Companion, discovered the disabled spacecraft of the Human inventor of Earth's warp drive, Zefram Cochrane. The entity brought the ship and its pilot, Cochrane, safely down to its homeworld. It rejuvenated the eighty-seven year-old inventor and terraformed the planetoid to sustain Cochrane's life.
In 2267, while transporting a terminally-ill Commissioner Nancy Hedford from Epsilon Canaris III to the USS Enterprise, the shuttlecraft Galileo was pulled off course by the Companion and forced to land on its homeworld. Cochrane had told the entity that he would die of loneliness if he didn't have contact with other Humans. The Companion merged with Commissioner Hedford, moments before her death, and Cochrane remained with her on the planetoid. James T. Kirk promised Cochrane that he would not tell Starfleet that they had discovered him or of what had happened on the planetoid. (TOS: "Metamorphosis"; ENT: "Broken Bow")
According to Startrek.com, Star Trek Concordance and Gold Key Comics Star Trek issue 49 "A Warp in Space", Gamma Canaris N was the name of the Companion's homeworld.   According to the "The Explored Galaxy" star chart, the Gamma Canaris region, was located between the Tholian Assembly and Romulus, both of which were identified in DS9: "Call to Arms" as Alpha Quadrant powers.
While "Metamorphosis" stated Cochrane disappeared "a hundred and fifty years ago" (or in 2117), it was established later on, in the Star Trek: Enterprise episode "Broken Bow", that Cochrane was still on Earth in 2119, at the opening ceremony of the Warp Five Complex.