Since civil registration of births in Ireland only started in 1864 the other bit of information you will need to know in order to search for church records (after you find where they lived- civil parish if not townland, not just the county) is the family's religion.
Civil registration of non-Catholic marriages started in 1845 so there's a chance, if the family weren't Catholic, that there might be a marriage registration for William's parents.
Often families, or people from the same area, settled near each other. Have you found any relatives nearby? If so, maybe there are clues in the census as to where they were from in Ireland.