No, they actually come from Mexico. The confusion arose due to the fact that they were introduced into central Europe by Turkish merchants.
Just to add to the confusion, because America had been discovered by explorers seeking an alternative route to India and the East, other nations named the bird assuming it was of Indian descent.
In France the turkey was called coq d'Inde, (now corrupted to dindon). In Italy, turkey was galle d'India, in Germany the name was indianische henn, while throughout the Ottoman Empire it was called the hindi.
Other tasty morsels of information like this can be found in Christmas Miscellany: Everything You Always Wanted to Know About Christmas, out this fall from Skyhorse Publishing.
But to keep you amused until then, why not take part in a festively-themed turkey shoot, by clicking this link?