There are several cemeteries in the Knoxville area that have the name Brown Cemetery. This one is located at 1055 Brown Road in south Knoxville.
The Knox County GIS website calls it Brown Cemetery IV on its map.
The website FindAGrave.com has a page for this cemetery, too. It lists 16 interments, with the oldest one being for Henry A. Brown. He was born on May 17, 1843 and died on April 6, 1846. The person with the earliest birth date listed is Mary Gossett Brown, who was born on October 21, 1812 and died on January 21, 1896.