The European Hegemony was an organization that existed in the aftermath of World War III. 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. (TNG: "Up The Long Ladder", "Attached")
A distress call used by the European Hegemony, circa 2123, was sent from the Ficus sector, and was received in 2365 by Starfleet. (TNG: "Up The Long Ladder")
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.