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 Ann(e) Johnston B 1838 +/- 2 yrs, somewhere in Ireland, with a father forenamed W(illiam), and mother El(????), I found 1 record, but NOT in Co. Monaghan. Also, the mother's forename on this record is listed as Ellen, not Eliza.
For Eb(enezer) Wilkin there are NO birth or marriage records at any date, anywhere in (transcribed surviving record) in the database.