It looks like Hosea Godfrey is the son of Daniel Godfrey, who was born about 1791 or 1792. In 1850 Oswego County, New York census there is a Hosey Godfrey, age 17, in the house of Daniel Godfrey. Hosea's brother, Alonzo named one of his sons, Miles, born in 1857.
Daniel was the son of Benjamin Godfrey and Sally Crosson, and Anna's grandfather, Thomas Crosson Godfrey was the son of the same Benjamin Godfrey, so, Hosea Godfrey married his first cousin once removed, Anna in Wisconsin, but can't find Hosea Godfrey in 1860 census. Hosey and Anna Godfrey are in 1870 Lancaster County, Nebraska census, and in 1900 Big Horn County, Wyoming census where the handwriting looks like F. M. Godfrey. All hese census records place Hosea's birth as between 1833 and 1835, but censuses can be wrong on ages by a few years, sometimes! Can't find them in 1880 census, though their children born around that time were born in Colorado.