That Michael and Elizabeth were married in Ballymore in 1862 suggests that she may have been from that parish. Ballymore registers go back to 1820s, so you may well be able to find her birth. But the first register was not filmed by NLI, and therefore is not online.
Actually, there is an excellent candidate for Elizabeth Furlong in the Lady's Island parish registers
Baptized ... December 1838, parents William Furlong and Anastasia Mythen, in Reedstown
https://registers.nli.ie/registers/vtls000634035#page/5/mode/1upThe Christian names William and Anastasia match with Elizabeth and Mathew's first daughter and second son, in good agreement with naming patterns
Reedstown is in the Tacumsin/Tacumshane area, bordering Our Lady's Island lake, see
https://www.townlands.ie/wexford/forth/tacumshin/tacumshin/reedstown/