Carthage is the county seat of Moore County, North Carolina, United States. The population was 2,775 at the 2020 census, up from 2,205 …Carthage is the county seat of Moore County, North Carolina, United States. The population was 2,775 at the 2020 census, up from 2,205 in 2010. Carthage is named after the ancient city in northern Africa.