What changed about the Greek-Turkish relationship that inspired Greece to rebel after being a part of the empire for so long?
The Greeks had rebelled prior to this, it just happened to be the first successful uprising. They were usually in sync with Italian (well, venetian) wars waged against the ottomans, with the exception of the orlov revolt during the russo-turkish wars of the late 18th century.
The fundamental differences were the scales of the revolts, as they tended to be rather regionalized prior, and lack of foreign support. The orlov revolt itself was supposed to have been aided by the Russians, but they never actually delivered their promised troops. Contrast this with the Greek revolution, where the Greeks received support from 3 great powers ( UK, Russia, France).
Another contributing factor would have been the growth of nationalism since the French Revolution, the received support from catherine the great of russia for greek nationalism, and the relative success of the Second Serbian Uprising in 1815, which was the first christian ottoman subject to receive concessions from the ottomans.
There is another matter which is important to note, and that is that the number of educated Greeks, and well to do Greeks, in the ottoman empire grew significantly in the 17th and 18th centuries (this is actually in contrast to how it was with the Serbian populace, as the majority of their elite came from austria-hungary, not really relevant, but I feel an interesting side note). This educated elite, as was the case in France and the 13 colonies) were crucial for garnering widespread support for the revolt.
Will provide sources when I'm home (there's about 10 here) sorry for typos and punctuation, on mobile.
Edit: just to add, it is obviously important to note that it was seizing the opportunity of the steady ottoman decline since the failed siege of Vienna.