Turquoise is named after a country where it is not mined.
Wikipedia calls the name a misnomer. The stone came from Persia and is not found in Turkey. Europeans named it after the route it traveled, not the place it was dug up.
The word is French. Etymonline gives the Old French phrase pierre turqueise, "Turkish stone." Turqueise is the feminine form of the adjective for "Turkish." Middle English had already borrowed forms such as turkeis by the late 14th century. The modern spelling came over from French in the 1560s and replaced them. Spanish turquesa and German Türkis carry the same idea.
Why Turkish? Etymonline says the stone was so called because it first reached Europe from Turkestan or through Ottoman lands. Wikipedia adds that it probably came to Europe through Turkey with other Silk Road novelties. Britannica agrees that the stone traveled to Europe through Turkey, which probably accounts for the name. In its turkey entry, Etymonline notes that 16th-century "Turkey" could vaguely mean Asia or the Ottoman lands.
The real sources are far older than the name. Wikipedia says the Nishapur workings in Iran and the mines of the Sinai Peninsula are the oldest known. Egypt used and mined turquoise from at least the First Dynasty, around 3000 BCE. Britannica calls the Sinai workings, begun before the 4th millennium BC, one of the first important hard-rock mining operations. Iran has been an important source for at least 2,000 years.
The people who mined it had their own word. Iranians called the stone pērōzah, "victory." Arabs later called it fayrūzah, pronounced fīrūzeh in modern Persian. None of that reached English, which kept the middlemen's label.
The color name followed the stone. Wikipedia dates the first English use of turquoise as a color name to 1573. Etymonline dates the color adjective to the 1570s and the stand-alone color noun to 1853.
Turquoise has company in this kind of mix-up. Etymonline says the American turkey got its name partly because Europeans identified it with the guinea fowl, a bird that had reached Europe through Turkish territory. Both names credit Turkey for something it did not produce.