Dawn,
You have given the middle child’s name as Rebedda. I have not come across such a name and assume it’s an error for Rebecca. The indexes on the Ulster Historical, Foundation website lists an Elizabeth Curry baptised 1839 and a Rebecca Curry in 1842 to parents James Curry and Jane Finlay. (Pay to view).
There was also a Susannah in 1845 and an Amelia in 1846 to parents of the same names, which you may want to investigate.
https://www.ancestryireland.com
Those records should tell you what denomination the family was. (I have a hunch it may be St Anne's Church of Ireland). That might be the church where James was baptised. Perhaps also his wife. I don’t see their marriage listed suggesting Jane was from a church whose records are either lost or not on-line. (Tradition was to marry in the bride’s church).
If you have definite evidence James & Jane were both born in Belfast then you could try searching for baptisms, but it’s worth bearing in mind that the city expanded dramatically in the 1800s and a huge proportion of the population came from outside the city. The population in 1800 was 20,000. By 1901 it was 350,000. Hundreds of thousands of people poured into the city from the surrounding counties all through the 1800s, to take advantage of the new jobs created by the shipyards, linen mills and related industries. So someone who married in Belfast c 1838, as in your case, may not necessarily have been born there.
Your next problem, is that a lot of the records are not on-line. Many Presbyterian records are not on-line and not all Church of Ireland records are either. No Methodist or most other minor denominations are on-line. The most complete collection is held in PRONI (the public record office) in Belfast but a personal visit is required to view them. And it’s shut at present due to Covid.
