Yeah that blew my mind too. Apparently the village of Montauk, MO and the state park were named after the Montaukett tribe.

https://web.archive.org/web/20160624071627/http://shsmo.org/manuscripts/ramsay/ramsay_dent.html

> Place name: Montauk

> Description: A village in the central part of Current Township; the second post office established in the county (1860). H.D. Kitchens was an early postmaster. Ephraim Bressie named it Montauk. Since 1927 it has been a state park. The large spring there is the head of Current River. One of the first mills in the county was built there by C.L. Stevenson in 1868. He sold to a Mrs. Hickman and her son in 1881. The locality has been known from the mill owners as Stevenson's Mill and Hickman's Mill, but the name Montauk has always been the official name for the village which was the first county seat. The name is an Indian name of uncertain meaning, according to Hodge. He states that it is the name given a tribe formerly from Long Island and related to the Indians of Massachusetts and Connecticut. Gannett says that it is a corruption of the Indian "minnawtawkit," meaning "island place," or "in the island country." By another authority said to mean "spirit" or "spirit tree." Montauk was first called Bressie's Spring, for Mr. Ephraim Bressie who came to the region in 1830 from Tennessee. (Postal Guide; HIST. DENT, pp. 567, 597; Holbrook; Hodge, p. 934; Gannett, p. 212; J. McDonald)

> Source: O'Brien, Anna. "Place Names Of Five Central Southern Counties of Missouri." M.A. thesis., University of Missouri-Columbia, 1939.

Apparently Ephraim Bressie chose the name. Unclear on whether any Montaukett were there, but apparently a Trail of Tears route went through Dent County, MO around the same time Montauk, MO was founded, so it’s possible. If no Montaukett were there maybe it was the member of another tribe or Ephraim Bressie’s way of honoring them and preserving their name.

https://www.salemmo.com/dent/about/index.php