I search the IFHF's online database @ ROOTSIRELAND.IE. It has abt 19 million B/M/D and other records for MOST Irish counties. Records before 1863 are increasingly sparse and almost non-existant before 1800.
For Mary Fleming B 1831 +/- 2 yrs, somewhere in Ireland, with the parents' names unknown, I found 58 records. Of these, 6 were in Co. Laois (Queen's). None of the 6 was in 1831 (exact).
Looks like you need to do more digging. Hope this helps. Good luck.