I can?t see any death in Ireland in the statutory records, for an Isaac Fleming on or around 26.7.1874. It?s not a common name and only 7 people of that name died anywhere in Ireland in the period 1864 to 1900. The only one in the Newtownards registration area died 12th Aug 1883, est year of birth 1839. So either the gravestone information contains an error, or Isaac died outside Ireland and was brought back to Newtownards for burial. (Many gravestones were erected years after the person had died and contain mistakes. Faded and weather worn inscriptions can also lead to transcription errors).
This Co Down site mentions a couple of Isaac Flemings living in Newtownards in the 1830s who might be of interest:
http://freepages.genealogy.rootsweb.ancestry.com/%7Erosdavies/SURNAMES/F/Fl.htm
Statutory birth and death registration began in Ireland in 1864. Non RC marriages were registered from April 1845. Prior to those years you are dependent on church records to find events. Many church records are lost or incomplete and not all that survive are on-line.
You don?t say what denomination the Fleming family were. I?d hazard a guess they were Presbyterian, though they might have been some other denomination eg Church of Ireland. As far as tracing Matthew?s baptism around 1754 is concerned, unfortunately the earliest surviving records in Newtownards for any denomination only date to 1821. There are none for the 1700s so it?s not going to be possible to trace him that way.
You believe he married, possibly to Eliza, in Belfast c 1780. Tradition was to marry in the bride?s church, so it?s her denomination that matters when searching for that event, as well as the baptisms of Lucy and Isaac (which may well be in a different church). There are several Presbyterian and Church of Ireland churches in Belfast whose records cover the years 1780 ? 1800. Copies are held in PRONI (the public record office) in Belfast. Your daughter may want to look at them when visiting. (Allow at least half a day).
I had a look in the Belfast street directories for the 1800s but did not see a Matthew Fleming listed.
http://www.proni.gov.uk/index/search_the_archives/street_directories.htm
PRONI have a record of an Isaac Fleming who was a special constable in 1812 ? 1816 (ref D46/1) but whether he is the Isaac you are interested in, I can?t say.
Sorry not to have come up with any information to help you but I can?t find anything in the on-line records. Some time may need to be spent on the records that are not on-line, in PRONI.