Multiple realities
(covers information from several alternate timelines)

The location of Gamma Hydra on a tactical situation monitor

The location of Gamma Hydra on a tactical intelligence analysis

Gamma Hydra was a planetary system in the Gamma Hydra sector of the Beta Quadrant. Located near both the Federation-Klingon border and the Romulan Neutral Zone, this system fell within the administrative area of Starbase 10. Additionally, the star was located six light years from a binary star system, where, in 2256, the first battle of the Federation-Klingon War occurred. (TOS: "The Deadly Years"; Star Trek II: The Wrath of Khan; DIS: "The Vulcan Hello")

Location

In 2366, the location of Gamma Hydra was labeled on a tactical intelligence analysis star chart. (TNG: "The Defector" okudagram)

In 2367, the location of Gamma Hydra was labeled on a tactical situation monitor in Captain Benjamin Maxwell's ready room on the USS Phoenix. (TNG: "The Wounded" okudagram)

In 2368, the location of Gamma Hydra was labeled on a tachyon detection grid. (TNG: "Redemption II" remastered okudagram graphic)

In 2366, in an alternate timeline, during the Federation-Klingon War, the location of Gamma Hydra was labeled on a tactical situation monitor in the ready room aboard the USS Enterprise-D. (TNG: "Yesterday's Enterprise" okudagram)

History

By 2256, an Andorian colony had been established in this system. It was considered vulnerable to Klingon attack should the USS Shenzhou fail to find a diplomatic solution to the standoff with the Sarcophagus prior to the Battle of the Binary Stars. (DIS: "The Vulcan Hello")

In the mid-23rd century, a Federation experimental colony was founded on Gamma Hydra IV. In 2267, a rogue comet intersected with the orbital path of Gamma Hydra IV. As the trail of the comet passed directly through the planet, it was irradiated by with residue left behind by the comet. This radiation affected the colony, resulting in a total loss of life. (TOS: "The Deadly Years")

In 2285, in the Kobayashi Maru exam, the USS Enterprise was on a training mission to Gamma Hydra. (Star Trek II: The Wrath of Khan)

This system's quadrant of origin is inferred based on the location of Sigma Nelvana's system as seen in the star charts appearing in the Star Trek: The Next Generation episode "The Defector" and in the Star Trek: Picard episode "Maps and Legends".
According to Star Trek: Star Charts (p. 56, "United Federation of Planets II") and Stellar Cartography: The Starfleet Reference Library ("Federation Historical Highlights, 2161-2385"), Gamma Hydra (spelled as "Gamma Hydrae" in these reference works) was a G-class star. The star's system was a destination on the major space lanes.

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