Oshkosh is a city in Garden County, Nebraska, United States. The population was 884 at the 2010 census. It is the county seat of Garde…Oshkosh is a city in Garden County, Nebraska, United States. The population was 884 at the 2010 census. It is the county seat of Garden County. Oshkosh was founded in the 1880s by cattlemen who found the surrounding area ideal for livestock grazing. It was named after the city of Oshkosh, Wisconsin. The first post office in Oshkosh was established in 1889.