I search the IFHF's online database @ ROOTSIRELAND.IE. It has abt 19 million B/M/D and other records for MOST Irish counties. Records before 1863 are increasingly scarce and almost non-existant prior to 1800. Early B/M/D events either weren't recorded, or the records haven't survived.
For Thomas Quinn B 1755 +/- 10 yrs, Co. Armagh, with the parents' names unknown, I found 0 records. When I broaden the search to include MOST Irish counties (except Clare, Carlow, Kerry, and parts of Dublin and Cork), I find 10 records.