No, poinsettias are native to Central America. The reason you see so many of them in Turkey is because they are a symbol of Turkish patriotism.
The myth of Turkish poinsettias began because they are the favorite flower of the founding father of Turkey, Mustafa Kemal Atatürk, who cultivated their growth across the country. Locals now commonly refer to them as the Ataturk flower (Ataturk cicegi).
The Aztecs, however, first called them cuetlaxochitl, meaning the flower that grows in residues or soil. In Mexico and Guatemala, they are known today as flor de Noche Buena, which translates as flowers of the holy night. They are also known in Central America as the Christmas Eve flower, which is in reference to the holy night being the one when Jesus was born.