Aldebaran III was the inhabited third planet in the Aldebaran system. This system, which was near to the Ficus sector, was located in the Alpha Quadrant. (TOS: "The Deadly Years"; TNG: "Up The Long Ladder", okudagram)
During the 2260s, Doctors Janet and Theodore Wallace performed experiments using various carbohydrate compounds to slow down the degeneration of plant life at the Federation outpost on this planet. (TOS: "The Deadly Years")
This planet's quadrant of origin is inferred based on the position of its star system as seen in the star charts appearing in the Star Trek: Discovery episode "Magic to Make the Sanest Man Go Mad" and in the Star Trek: Picard episode "Maps and Legends".
According to StarTrek.com, Aldebaran III was a long-time Federation member. This website, and the Star Trek Encyclopedia (4th ed., vol. 1, p. 16) – in an article about the Aldebaran system – treated Aldebaron III and Aldebaran III as the same planet.