Armstrong

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 that do are on-line. In your case, in the 1700s, I would expect Armstrongs in Fermanagh  to have been Church of Ireland (Anglican). Likely descended from settlers from the Scottish borders who arrived in Ireland in the early 1600s.

RC records are mostly on-line on the nli site:  

https://www.nli.ie/en/family-history-introduction.aspx

For other denominations, the churches usually hold the originals but 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:

https://www.nidirect.gov.uk/publications/proni-guide-church-records

Brookborough is in the parish of Aghavea. Their Church of Ireland baptism, burial and marriage records only start in 1815 so you won’t find a record of your couple unless they were buried in the area, post 1815. Armstrong is about the second most common surname in Co. Fermanagh so, if you do find any likely records anywhere, care will be needed to ensure it is the right family. In the 1901 census of the county there were 54 people named James Armstrong. It will have been equally common in the late 1700s.

Researching in Ireland in the 1700s is very hard going due to the general lack of records. If you don’t know where they lived it’s a needle in a haystack. Ideally you need to know the person’s exact denomination and the townland or parish they lived in to have any chance of finding them, and even then there may not be any records for that location.

Possibly DNA testing may be a way of matching with others who have additional information about where the family originate. Family Tree DNA reportedly has more people with Ulster roots than any other company. That obviously increases the chances of finding a match. You might want to try them or, if you have already tested, you can transfer your results to them for no fee.

The North of Ireland Family History Society is running an Ulster DNA project in conjunction with FTDNA and can offer testing kits at a reduced price.  http://www.nifhs.org (Go to DNA project on the website).

Thanks so much for the information. Hope your Christmas and New Year are festive and safe.