The European Hegemony was a somewhat loose alliance formed in the early 22nd century. Jean-Luc Picard described the Hegemony as one of the first steps toward a United Earth Government, which was founded in 2150. In the 24th century the European Hegemony did no longer exist. (TNG: "Up The Long Ladder", "Attached")
A distress call in common use by the European Hegemony between 2123 and 2190 was sent from the Ficus sector, and was received in 2365 by Starfleet. (TNG: "Up The Long Ladder")
- The fact that Bringloidi send this signal may imply they were formally citizens of the Hegemony. Since they seemed mainly of Irish descent and their ship was launched out of Baikonur Cosmodrome, Ireland and Kazakhstan might have been members. Furthermore the Spanish name of their ship might imply Spain being a member. However all this is very circumstantial evidence and should not be treated as more than educated guesses.
Background
It is conceivable that the Hegemony was an outgrowth or expansion of the European Union, which was formed in the late 20th century. Furthermore, it remains unclear which states were member of the Hegemony, and how it is related to the future European Alliance. Despite it being described as one of the first steps towards earth government, it apparently survived the incorporation of that government at least in some form, as evidenced by the date of 2290.