Brewster County Courthouse - The Brewster County Courthouse is located in Alpine Texas which is in far west TX. Brewster County is a county located in the western part of the U.S. state of Texas. The latest population was 9,232 per the 2010 census. Its county seat and only city is Alpine. The county is named for Colonel Henry Percy Brewster, a Secretary of War for the Republic of Texas. One thing that is the color of the courthouse is red with a gazebo next to it. Photo © copyright by Bee Creek Photography - Tod Grubbs / Cynthia Hestand.