They resided in Ohio. There is an Olive and J.M. Palmer in the 1880 Census of Bridgewater, Williams Co., OH. The transcription of this census at FamilySearch.com may have errors. It says that J. M. Palmer is a farmer, age 55, and born in NY. Both his parents were born in NY, too. Olive is 54 and it says she is born in OH. Her parents' birth states were either not readable or not listed. However, the two sons listed, Harley (11, OH, NY, NY) and Francis (9, OH, NY, NY), both show that their mother was born in NY. If Olive is not their mother, this would be okay. If Olive is their mother, then there is a mistake here. Olive should be listed as born in NY.
From Ronny Bodine:
The Ohio death record of Sylvia Hutchison, who died 3 Feb 1927, records her birth on 18 Sept 1840 in Williams County, Ohio as the daughter of James Michael Palmer and Olive Bordine.
1850 Bridgewater, Williams Co., OH: James PARMER [sic] 26 NY farmer, Ollive 25 NY, Sylvia 3 OH.
1860 Bridgewater, Williams Co., OH: J. M. PALMER 35 NY farmer, Olive 33 NY, John 13 OH, Harriet 9 OH, Margaret 3 OH, George 2 OH.
1880 Bridgewater, Williams Co., OH: J. M. PALMER 55 NY farmer, Olive 54 NY wife, Harley 11 OH son, Francis 9 OH son.
James and Olive Palmer appear to have died between the June 1880 and June 1900 censuses.