So, the name similarity between the group of ethnicities and the color is evident. I would believe it to be a coincidence if it wasn't for the fact I have noticed the turquoise color being used various times in Turanist iconography.
So, what gives? Where does this tradition come from?
You are right, it's not a coincidence. Turquoise, like many other words in English, is derived from Old French pierres turqueise "turkish stones". The name for the stone in Italian "turquino" , Portuguese "turqueza", Spanish "turquesa" , German "türkis", Dutch "turkoois", Danish "turkis", Swedish "turkos" all point to a shared etymology.
The Oxford English Dictionary quotes the Travels of Marco Polo where he mentions pierres qui s'appellent turquesses ("stones which are called turquesses"). So, the association of the stone with assumed origin in Turkestan seems to already be present in the 1300s.
OED also provides a quote of an English translation from 1398 of De Proprietabus Rerum
De Turtogis. Turtogis that hatte Turkeis also is a ȝelow white stone and haþ þat name of the contrey of Turkeis. Þis stone kepeþ and saueþ þe siȝt and bredeþ gladnes and comforte.
in modern English that would say: the Turtogis. Turtogis that is also [called] Turkeis is a yellow-white stone and has that name of the country Turkeis. This stone keepeth and saveth the sight and breedeth gladness and comfort.
so, in 1398 Bartholomew de Glanville is explicitly saying Turqoise stone gets its name from Turkish country.
source: https://www-oed-com/view/Entry/207762?redirectedFrom=turquoise