Clermont is a city in Fayette County, Iowa, United States. The population was 586 at the 2020 census. Clermont is home to Montauk, the man…Clermont is a city in Fayette County, Iowa, United States. The population was 586 at the 2020 census. Clermont is home to Montauk, the mansion of former Iowa governor William Larrabee, along with much historic architecture. Scenes for the movie The Straight Story were filmed here.