I searched the IFHF's online database at ROOTSIRELAND.IE. It has abt 20 M church, civil, census, etc B/M/D records for MOST of Ireland, including Tyrone. Records before 1863 are increasingly sparse - almost non-existant before 1800.
For Peter Quinn B 1750 +/- 5 yrs, somewhere in Ireland, I found 0 records. It is most likely that the event was never recorded, or simply hasn't survived.