Now, you or anyone else can certainly spell it "Türkiye" instead of "Turkey", but a) you risk confusing people (which you may or may not care about), and b) typing "ü" is awkward on many (most? all?) English-layout keyboards. I suppose you could also spell it "Turkiye", which I suppose is closer to what the Turkish government wants, but is still "incorrect".
At any rate, I personally see little reason to change unless popular usage overwhelmingly changes. At least in the US, popular usage (which influences dictionaries) dictates English spelling, not governments.