No it was not claimed by the French
A man named Samuel De Champlain claimed New York.
Samuel de Champlain
New Hampshire was claimed by Vermont, Massachusetts, and New York.
The French claimed New Franc, hoping they could make a colony. They also wanted more fur for business.
In 10,000 BC the Native Americans. Then the Europeans discovered New York in 1524. (led by the French) Then in 1609 it was claimed by the Dutch. In 1664 the Europeans named it New York after the Duke of York. It was a major trading in the thirteen colonies. I think.
This area was first claimed by Holland.
New York and New Hampshire claimed the Hudson River.
England, a clue of that is because New York state is in New England
Henry Hudson
"Born in New York" in French is spelled as "né à New York."
Ellis Island
Dutch, England, France