I search the IFHF's online database at ROOTSIRELAND.IE. It has abt 20 M church, civil, census, etc records for MOST Irish counties. Records before 1863 are increasingly sparse - almost non-existant before 1800.
For Peter McDermott B 1876 +/- 1 yr, somewhere in Ireland, with the parents' names unknown, I found 10 records. NONE were in Co. Monaghan. This county's records were only added to the database recently. There may be more at some time. Suggest you visit the IFHF database, register (free), browse (also free), and opt to be notified as records are added.