I searched the Irish Family History Foundation's online database at ROOTSIRELAND.IE. It has abt 20 M B/M/D, civil, census, etc records for most Irish counties. Records before 1863 are increasingly scarce - and almost non-existant before 1800.
For Win(ifred) Conroy B 1888 +/- 5 yrs, somewhere in Ireland, with the parents' names unknown, I found 15 B records. None are in Co. Cork. Four have a father forenamed Mich(ael). None have a mother forenamed Cat(herine).
For Jul(ia) Conroy B ?, somewhere in Ireland, with a father fornamed Mich(ael), I found 8 records, with one in Co. Cork.