This page contains information regarding Star Trek: Strange New Worlds, and thus may contain spoilers.
The Gorn Hegemony was the governmental body of the Gorn, a reptilian species first encountered by the United Federation of Planets in the mid-23rd century. The five systems were part of their domain. (ENT: "Bound")
The Orion privateer Harrad-Sar had dealings with the Hegemony prior to 2154, but declined to elaborate to the crew of Enterprise NX-01, merely saying "The less said about them, the better." (ENT: "Bound")
In the 2230s or 2240s, the Gorn attacked and captured the SS Puget Sound, a Federation colony ship, and deposited its crew and passengers on a Gorn breeding planet to be hunted for sport or eaten alive by their hatchlings. La'an Noonien-Singh was the sole survivor of the incident, and related the story to Starfleet; this was the first documented encounter with the Gorn, although it was not considered an official first contact. (SNW: "Strange New Worlds")
In 2259, the Gorn attacked and massacred the colony of Finibus III, and then attacked the USS Enterprise when it arrived in the system. During the subsequent battle, the Enterprise was badly damaged and lost seven crew members, but managed to destroy three Gorn hunters before evading a destroyer using the system's black hole as cover. With the Gorn confirmed to be raiding targets in Federation space, Captain Christopher Pike resolved to be more prepared for the next hostile encounter. (SNW: "Memento Mori")
Official first contact finally took place eight years later, in 2267. This encounter with the Hegemony was also hostile, as the Gorn were claiming Cestus III – a world which was then occupied by a Federation settlement – as their own. Despite this, the two powers had resolved the conflict over Cestus III by the 2370s, when a Human settlement thrived there. (TOS: "Arena"; DS9: "Family Business")
The name of this government was revealed in "Bound", although there was apparently no official UFP contact with the Gorn until Kirk's era. Federation-Gorn relations were described, in the Star Trek Encyclopedia, as "cordial" during the 2370s. The name "Gorn Hegemony" had been used in non-canon books and games for years before this episode was written, however.
In the novel Articles of the Federation, one of the characters says that the crew of the USS Enterprise convinced the Gorn to join the Federation Alliance and fight against the Dominion in the war. This is from the graphic novel "The Gorn Crisis" by Kevin J. Anderson and illustrated by Igor Kordey. But in the game manual for Star Trek: Bridge Commander, it is stated that the Gorn sided with the Dominion. The Star Trek: Destiny novel Mere Mortals by David Mack establishes that the Gorn Hegemony had an embassy to the Federation located in Berlin and its head of government was the Gorn Imperator. In A Singular Destiny, the Gorn allied themselves with the Romulan Star Empire, the Tholians, and others antagonistic towards the Federation and Klingons.
In Star Trek Online, by the early 25th century, the Klingons had conquered the Hegemony after learning that their leaders had been replaced by Species 8472 infiltrators, and absorbed the Gorn into the Empire. The storyline of the game also states that a monarch, King Slathis, was the head of the government.
The twenty-fourth issue of the Star Trek: Ongoing comic book series features a group of Gorn who had rebelled and settled on the planet Parthenon 559 wanting to live in peace. But because of the previous encounter as mentioned above, the Humans who came to mine the planet reacted with hostility. Kirk resolved everything and forced the Humans off of the planet after learning more about the Gorn in the process. The issue also potentially provides another origin story for the Gorn Hegemony.