They traded with Spain (even thought they didn't much like the Spanish) to get wine and other goods, and the Spanish had occupied Mexico and gotten turkeys and other foods from there.

Most crops such as corn, pumpkins, and turkeys associated with American Indians in popular culture weren't originally indigenous to the United States but spread north from Mexico and Central America in ancient times.

I think Muslims were indeed present in the Americas prior to the 1800s. IIRC Sir Francis Drake had liberated some Barbary corsairs from Spanish prisons in South America on his raids against Spain. Some Americans might be descended from an early Turkish or Morisco presence but this may not be supported by recent molecular studies: https://en.wikipedia.org/wiki/Melungeon