Not according to the Smithsonian - and they should know because they have the paperwork.
One of their series on American myths just before the Bicentenial was on Thanksgiving. It was signed into our holiday schedule by Abraham Lincoln under pressure from one of those periodic Christian revivals we have in this country.
The Thanksgiving story was made up completely out of whole cloth, just as the Pilgrim's costumes were, and their hunting wild turkeys with a blunderbuss - a splatter gun used as a short range anti-boarding weapon on ships. I think the costumes were copied from inside the lid of a box of Dutch Master cigars.
The only actual record of a "feast" in the Plymouth colony was a note by one of the leaders lamenting an out-of-control beer bash that lasted days and ate up much of the winter supplies. He mentioned that some Indians came by when they smelled the food and were sent out to get deer. The sanctimonious thanks to God part was completely made up in Lincoln's day.
If turkeys were eaten - completely unconfirmed - they were likely Mexican turkeys brought over on the ship. By time of the Pilgrims' voyage Mexican turkeys had completely displaced the peacock as festival bird in Europe and it certainly would have been within reason for them to bring some.
Wild turkeys are smart, fast and extremely difficult to bag. Many enthusiastic turkey hunters with modern gear have never been able to actually get one - and you certainly aren't likely to get one with a blunderbuss.