Domesticated South American turkeys were brought to Europe by the early Spanish explorers. The original Pilgrims brought some of those turkeys back to North American when they came to settle. Some got into the wild and now all wild US turkeys have some of the DNA from those ‘European’ turkeys. The Indians didn’t introduce the Pilgrims to turkeys...the Pilgrims already had them.
That was after 1620, but there were turkeys in England before 1492, and turkey corn as well. They were called turkeys because it was thought they came from Turkey. Too simple, probably.