The Dubbel/Triple/Quad designations are totally separate from the Trappist label.
The Trappists are an official Catholic religious order that follow the rules of St. Benedict, one of which states "for then are they monks in truth, if they live by the work of their hands". The monasteries all make goods, most commonly beer and cheese, that they sell in order to fund themselves. Several of the monasteries form the International Trappist Association in order to prevent anyone who's not actually a part of the order from using the name Trappist on their products.
But Trappist ale isn't really a style, it's a commercial seal like "Organic". Both Dubbel and Tripel originated as names of specific beers brewed by Trappists at the Westmalle abbey. Dubbel coming to use in the 1850s and Tripel (1950s) being a renamed and slightly hoppier version of what the monks used to call Superbier (1930s). Quadrupel is a brand of La Trappe made at the Koningshoeven abbey, which was probably introduced some time in the 1990's. Koningshoeven Also used to market a beer named Enkel, but stopped production in 2000.
In all cases the popularity of the beers led to imitation by other brewers which sort of turned their brand names into loose stylistic designations. It'd be like if people started imitating Stone's Arrogant Bastard and started producing their own Arrogant ale.