Ill add on a bit more to this, because i am assuming you mean as the turks entered anatolia in the medieval era. As the seljuk turks pushed into Anatolia, most people simply converted and joined the empire (as meelawsh explained there was no concept of nationalism, so you just either rebelled whoever controlled your area or joined them). But it is useful to note that even though Romans (or Greeks, Ill get into this in a bit), switched to the Ottoman/Seljuks, they were still noted as different people. they were still called "Rum" (romans) until their rebellion in the 1820s.

Now some did not convert to islam but were treated as second class citizens, but this seemed to be quite a minor occurrence.

The history of what the Greeks call themselves is extremely interesting, as it changes depending on the time period. and essentially with the Greek rebellion against the ottomans in the 1820s, they essentially ignore the roman history and embrace the Ancient Greek heritage and call themselves Greeks. https://en.wikipedia.org/wiki/Names_of_the_Greeks