According to Wikipedia, Natchaug is a Nipmuc Native American word meaning "land between the rivers".
That said, the name of the forest does not mean the Nipmucs lived there.
The Nipmuc or Nipmuck people are an Indigenous people of the Northeastern Woodlands, who historically spoke an Eastern Algonquian language.
Their historic territory Nippenet, "the freshwater pond place," is in central Massachusetts and nearby parts of Connecticut and Rhode Island.
As Connecticut's official government website puts it, the forest was established as a state forest in 1917, and contains the remains of American Civil War, and Union Army General, Nathaniel Lyon's birthplace.
He was allegedly the first general killed in the American Civil War.