This first encounter with the Hegemony was 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 did indeed thrive there. (TOS: "Arena"; DS9: "Family Business")
Prior to actual contact there was little known by Earth of the Gorn. Among the minor bits of knowledge that did exist was that, according to Orion privateer Harrad-Sar in 2154, the Gorn Hegemony brewed "the finest Meridor in the five systems". (ENT: "Bound")
The name of this government was revealed in "Bound", although there was apparently no official UFP contact with the Gorn until Kirk's era. The name "Gorn Hegemony" had been used in non-canon books and video games for years before this episode was written, however. Since the full government name went unmentioned in later appearances, it is unknown if, or in what form, the Gorn Hegemony exists in the 24th century.
Federation-Gorn relations were described, in the Star Trek Encyclopedia, as "cordial" during the 2370s. While some references tell us that there was a settlement reached over the Cestus colony, there is no specific evidence from any canon source stating how friendly the Federation and the Hegemony might have become.
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 has an embassy to the Federation located in Berlin and its head of government is the Gorn Imperator. In A Singular Destiny, the Gorn ally 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 have conquered the Hegemony and absorbed the Gorn into the Empire. The storyline of the game also states that a monarch, King Slathis, is the head of the government.