Statutory recording of births only started in Ireland in 1864 so you won’t find a birth cert for William. You might find his baptism though. To do that you would need to search parish records. They are not on-line. Most have been copied and can be viewed at PRONI, Belfast. Lisburn is in the parish of Blaris. If you don't know his exact denomination, I’d be inclined to start with Presbyterian first, as that would be my guess.