John and Annie married in St Anne's Church of Ireland, but whether that was their actual denomination I'm not sure. Later generations were Presbyterian but that doesn't necessarily mean anything.
Tradition was to marry in the bride’s church, after which she would attend her husband’s. So it is possible that if they married in St Anne’s then Annie was Church of Ireland until that time. (If you have no other leads, then you might want to start your research on that basis).
You say Annie came from near Newry. Newry Church of Ireland records in PRONI are as follows (this is all on the website Aghadowey referred you to):
Newry (formerly St Patrick’s and then St Mary’s) (Dromore diocese)
Baptisms, 1804-14, 1817, 1819 and 1822-94; marriages 1784-1963; burials, 1824-76; index to baptisms, marriages and burials, 1784-c.1910; vestry minutes, 1775-1948; select vestry minutes, 1877-1909; registers of vestrymen, 1870-1935; preachers’ books, 1878-1957; registers of pews occupied, 1886-1926, and of subscribers, 1886-1959. MIC1/46-48; D2034
They are not computerised. They are on microfilm and if you don’t know the date, you will need to wind through all the years you think applicable. Watch out for possible siblings births (to the same parents) as well as Annie’s
Since you will have the film out anyway, you might look for John’s baptism too. If he’s not there, then I would try the various Presbyterian records for the area, as well as any other denomination you think might be possible. There are 4 Presbyterian churches within Newry parish but only 1 has records covering the years you are interested in. The others have either lost their early records or were not open in the 1830s and 1840s. For Presbyterians in Ireland, which church you attend is a matter of personal preference, and is not dictated by parish boundaries, and may not be the nearest church to where you live.
If you don’t find the baptisms in the records suggested, you will need to widen the search to adjacent parishes (for Church of Ireland) and churches (for other denominations). And there is the possibility that the family attended a church whose records for that period are lost.