Bogue is a city in Graham County, Kansas, United States. As of the 2020 census, the population of the city was 155. Bogue is located on K-18…Bogue is a city in Graham County, Kansas, United States. As of the 2020 census, the population of the city was 155. Bogue is located on K-18, south of U.S. Route 24, on the South Fork Solomon River.