Tucumcari is a city in and the county seat of Quay County, New Mexico, United States. The population was 5,278 at the 2020 census. Tucum…Tucumcari is a city in and the county seat of Quay County, New Mexico, United States. The population was 5,278 at the 2020 census. Tucumcari was founded in 1901, two years before Quay County was established.