The records you are looking for are before the start of statutory birth and marriage registration. You would therefore need to rely on church records, where they survive. Only a handful of churches records in Belfast for that era are on-line.
You would need to know the denomination and the church(es) where the marriage and baptisms took place. If you knew that, you could then look up the records in PRONI, Belfast. But if you don’t know that, then it’s a needle in a haystack. (There were over 100 churches in Belfast in the 1830s).