Matthew,
John Gilmore seems to have resided in Priest Town since the 1860s. He is there in 1863 in Griffiths Valuation and he died there in 1894. John & Rose married in 1851 at which time his townland was Ballyquintin. That’s a mile or two south of Portaferry, at the tip of the Ards peninsula. Not very far from Priest Town.
Statutory birth registration started in Ireland in 1864. I found 2 births for children from your couple:
Mary Anne Gilmore 27.8.1864
Patrick Gilmour 12.7.1867
https://civilrecords.irishgenealogy.ie/churchrecords/images/birth_returns/births_1864/03609/2331425.pdf
https://civilrecords.irishgenealogy.ie/churchrecords/images/birth_returns/births_1867/03478/2277918.pdf
The family farmed in Priest Town, near Downpatrick. John died there in 1894:
https://civilrecords.irishgenealogy.ie/churchrecords/images/deaths_returns/deaths_1894/05952/4691729.pdf
Rose died there in 1900:
https://civilrecords.irishgenealogy.ie/churchrecords/images/deaths_returns/deaths_1900/05775/4632622.pdf
Son William John was the informant. Here he is in the 1901 census:
http://www.census.nationalarchives.ie/pages/1901/Down/Ards_Upper/Priestown/1231887/
There was another Gilmore family nearby who presumably are related:
http://www.census.nationalarchives.ie/pages/1901/Down/Ards_Upper/Priestown/1231883/
Griffiths Valuation lists John Gilmore on plot 6a in Priest Town which was a 23 acre farm. That property today still looks to be a farm. It’s up a dead end lane off the Marlfield Rd, a mile or two north of Portaferry.
Letters of Administration (with the Will annexed) of the personal estate of John Gilmore late of Priestown County Down Farmer who died 11 October 1894 at same place were granted at Belfast to William John Gilmore of Priestown Farmer one of the Residuary Legatees.
The above will is on-line on the PRONI wills site. It mentions sons William John, John, Robert & Patrick as well as his wife.
Probate of the Will of Hugh Gilmore late of Priestown Portaferry County Down Farmer who died 27 March 1902 granted at Belfast to John M'Convey and William Breen Farmers.
Probate of the Will of Rose Gilmore late of Priestown Portaferry County Down Widow who died 20 September 1907 granted at Belfast to John McConvey Farmer and Rebecca Breen Spinster.
Above wills are also on-line.