Chartreuse is a color named after a drink, which was named after a monastery, which was named after a range of mountains.
The mountains come first. Etymonline says the Carthusian order and its mother house take their name from the Chartreuse mountain group in the French Alps, where the first monastery was built. St. Bruno founded the order in the 1080s at Chartreux, a village in the Dauphiné. The monastery became la Grande Chartreuse, and its monks the Carthusians, from Latin Cartusianus.
The drink came centuries later. Wikipedia says that in 1605 Marshal François Hannibal d'Estrées gave the Carthusian monks at Vauvert an alchemical manuscript. It held a recipe for an "elixir of long life." The monks began making a liqueur from it in 1737. In 1764 they adapted the recipe into the Elixir Végétal de la Grande Chartreuse. The milder green Chartreuse and the sweeter yellow followed in 1840.
The formula reportedly uses about 130 herbs, plants and flowers. Britannica describes the drink as distilled by the monks of la Grande Chartreuse, the order's motherhouse near Grenoble. Only two monks prepare the herbal mix and know it in full. It has survived two expulsions from France, in 1793 and 1903. After the second, the monks took the recipe to Tarragona in Catalonia.
English borrowed the liqueur's name in 1866, according to Etymonline. The color followed in 1884. Etymonline describes it as the pale apple-green of the finest version of the drink. Wikipedia puts the direction plainly: the color takes its name from the liqueur, not the reverse.
The same place name hides in English in disguise. London's Charterhouse school was built on the site of a Carthusian monastery. Etymonline calls its name a folk-etymology alteration of French chartreux. English speakers heard "charter house" in a French word for the monks.
A cat breed carries the name as well. A legend links the Chartreux to the monks. Wikipedia notes, though, that in 1972 the prior of the Grande Chartreuse said its archives held no record of monks keeping such cats.