(covers information from several alternate timelines)
Belgium was a country in Western Europe, on Earth. It was an European Alliance member state. Cities included Brussels, Antwerp, and Waterloo. The flag of Belgium was three vertical bands of black, yellow, and red. (ENT: "First Flight")
This region, like Armenia, was identified by Captain James T. Kirk in 2267 as a vulnerable way-point situated on the routes of many invading armies in Earth's history. (TOS: "Errand of Mercy")
In the alternate timeline 20th century created by Vosk and the Na'kuhl, Vladimir Lenin was assassinated and Adolf Hitler was able to focus on the West. He took France, and then Belgium and the Netherlands quickly fell. (ENT: "Storm Front, Part II")
In 2032, Belgium took part in the Ares IV mission. (VOY: "One Small Step")
In 2143, the Ariane 1 assignment patch, which was on display in the 602 Club, bore the flag of ESA member Belgium. (ENT: "First Flight")
In 2325, Devinoni Ral was born there. (TNG: "The Price")
In 2372, Antwerp was targeted by the Dominion. A changeling infiltrator detonated an explosive device there. (DS9: "Homefront")