I search the IFHF's online database at ROOTSIRELAND.IE. It has abt 19 million B/M/D and other records for MOST Irish counties.
For Pat(rick) Lyons B 1858 +/- 1 yr, Co. Tipperary, with a father forenamed John and mother named Margaret Quinn, I found 1 record in 1857. A view of the detailed record (usually for a nominal fee) SHOULD let you determine if this record is for your ancestor.
For Nora O'Neil B 1862 +/- 5 yrs, somewhere in Ireland, with the parents' names not used, I found 0 B records. Note - B/M/D records for Cork (west) and Cork City haven't been transcribed yet.
Suggest you visit the database, register (free), browse, explore, and wait until these records are transcribed and added to the database. The IFHF will inform you as additions occur.