They were already controlling Greece and were very close to Turkey. Turkey was also newly founded and very weak. So they could have defeated Turkey in a relatively short time. Why didn't they choose that path?
A few, simple reasons:
Turkey had agreed to sell Germany almost all the Chromite ore that they could muster(at quite high prices actually), in return for a non-agression treaty, which was signed in 1941.
Nazi Germany really, really liked Chromite, as it was vital for military production. Invading Turkey, taking over the mining operations and setting up a transportation path back home would simply take way too long.
Ismet Inonu, the president at that time, kept talking to both Churchill and Hitler, reassuring them of Turkey's neutrality every chance he got. Churchill did almost convince him to launch a major ground offensive on Germany though.
Germany likely did not want Anatolia and its precious mining facilities to be war zones between themselves and the Soviets, who were pretty much waiting around Turkey's eastern border, Kars.
Large Laz families, Turkmen, Kurdish and Circassian tribes, all have quite a large history of incredibly stubborn resistance, and Turkish Armed Forces were already experts at lengthy, well-organized guerilla warfare, in addition to being almost inhumanly patient at wars of attrition. An offensive pushing into that death trap full of nothing but mountains and long, hot steppes was simply not worth it, for anyone.
Veterans of the Turkestan Legion of Nazi Germany did write about them pressuring the German military into peace with Turkey as well; but given the minor size and strength of the Legion, that probably only did have very little impact.