King Willem I of the Netherlands (the Southern provinces / Belgium included) had the Royal Mews constructed in Wavre. And the Prince of Orange had the Écuries Royales in the park of the Palais du Prince d'Orange in the centre of Brussels. Part of the collection of carriages of Willem I and his son were confiscated and used by the new monarchy. Especially King Leopold II added a few gala berlines.
But the Kings Albert I and Leopold III had no special interest in it. Then World War II came, and the regency of Prince Charles. When King Baudouin started his Reign, he decided to hand over the Écuries Royales to the Academies' Palace, housed in the former Palais du Prince d'Orange. (This complex belonged to said palace). Without functioning Royal Mews in the centre of Brussels, King Baudouin ended the use of horses and carriages by the royal family.
There is still an Honorary Mounted Escort by the Federal Police, but these have nothing to do with the Royal Mews: the horses and the police officers are housed and trained in a former Army Barracks in Etterbeek (a municipality in Brussels agglomeration).