The word race presents problems. The question seem to suggest whether the Greeks of today share ancestors with the Turks of today?
Before we answer, we must look at the history of these two nations. |The Greeks had a long history. But the Greeks in the Helenic peninsula has had ancient Greeks , Romans, Scythians , Persians, South Slavs, Alans , and some germanic tribes settling amongst them. Alexander took over the Persian Empire, then the Eastern Roman Empire became Greek. The people of Trebisond Armenia, Kurds and celtic tribes (the Galatos) all marrying each other and adopting each other cultures.
Then came the turks who added themselves to the mix. One may say there are more Greek ancestors in many Turks today then the other way round through conversion and marriage. before 1924, there were Turks who spoke "Roman/Greek" . There is an interesting french traveller in the Napoleonic era who remark that the Turkish Pasha in Morea had Greek Wives and spoke to each other in Greek. All his descendants would have considered themselves TURK and has left Greece for Turkey by the treaty of Lausanne.
So are they of the same race? If you put a person from Ankara and one from Athens and dress them the same and ask both Turks and Greeks to say which one is which, if the y cant tell the difference then they are the same race.