You mention that John & Margaret married in Arclintagh in 1839. The tithes list a James Mawhinney farming there in 1830. So that might be Margaret’s family. There were only 15 farms there so not very big. He had 12 acres. Mawhinney/McWhinnie is the same name, obviously.
No Morrisons in that townland but there are some elsewhere in the parish. (People usually married close to home). McWhinnie (etc) is a rare name in Monaghan. By 1901 there were none in the county, so the chances of this being your family seem pretty high to me. I checked Griffiths Valuation for 1860 but there were no McWhinnie families in Arclintagh then. They appear to have either died out or migrated. Arclintagh is 145 acres of agricultural land. In 1901 there were 8 homes there and a total population of 21.
Morrison is a bit more common. In 1901 there were 37 in the county. Mostly Presbyterian, but 8 were Church of Ireland.
10 Morrison households in Clontibret in 1860:
Both Morrison & McWhinnie are Scottish names. That, combined with being in Monaghan suggests they are probably descended from Scots who settled there at the time of the Plantation. So you are probably looking at Presbyterian families, though they could be Church of Ireland. Methodism (which I see features in your post) had barely got going as a separate denomination in Ireland in the 1830s, and most Methodists were still using the Church of Ireland for baptisms & marriages then.
Statutory birth, death and marriage registration (in some jurisdictions called Vital Records) only started in Ireland in 1864, save for non RC marriages which were recorded from 1845 onwards. So you probably won’t find statutory birth, death or marriage certificates in Ireland for this family. For earlier years you usually need to rely on church records, where they exist. Ideally you need to know the precise denomination and have some idea of where the person was born in order to search the correct records. Not all churches have records for that period and not all are on-line.
RC records are mostly on-line on the nli site:
For other denominations, the churches usually hold the originals but for Monaghan and Ulster generally there are also copies in PRONI, the public record office, in Belfast. A personal visit is required to access them. Access to the records there is free. This link explains what records exist, parish by parish:
If you are unable to go yourself, you could employ a researcher. Researchers in the PRONI area: http://sgni.net
There are 2 Church of Ireland churches in Clontibret parish (Clontibret & Broomfield). Neither has any records for the years you need. There were apparently destroyed in the 1922 fire. 4 possible Presbyterian churches. 1stClontibret has records from 1825, 2nd from 1858, 1st Monaghan 1821 (with some gaps) & 2nd Monaghan or Ballyalbany start in 1802. Copes of them all in PRONI.