Your Martin was born around 1841. Birth registration didn’t start in Ireland till 1864. Prior to that you have to rely on church baptism records. To do that you need to know the person’s exact denomination and the likely church they attended. Not all churches have records for the 1840s and of those that do, not all are on-line.
As far as Co. Fermanagh is concerned, most RC parish records are on-line free on the National Library site:
For other denominations the main repository (for Ulster) is PRONI – the public record office – in Belfast. You haven’t said what denomination Martin was though statistically, he is likely to have been either Methodist or Church of Ireland. (Armstrong is a very common name in Fermanagh, being descendants of folk who moved there from the Scottish borders in the early 1600s. Most Armstrongs in the county are one of those 2 denominations, and prior to Methodism starting as a separate denomination around 1820, they’d have all been Church of Ireland).
The main Church of Ireland church in Enniskillen is St McCartan’s. Many of its records were lost in the 1922 fire. According to the PRONI catalogue this is what remains:
[Earliest registers destroyed in Dublin]
Extracts of baptisms, 1667-1789, marriages, 1668- 1794, and burials, 1667-1781; burials, 1879-1907 and 1941-50;vestry minutes, 1731-1920; copy deeds, 1796-9; select vestry minutes, 1871-80; register of church members, 1871 and 1946-50; preachers’ book,
1895-1928. Extracts from baptism, marriage and burial registers, 1666-1826.
Printed copy of Old Enniskillen Vestry Book, with extracts of births, marriages and deaths, 1666-c.1797. Extracts from vestry minutes, 1666-1912, which include some baptism, marriage and burial entries.
There is a second Church of Ireland church in the parish at Tempo, with records as follows:
Baptisms, 1836-1954; marriages, 1837-45; burials, 1837-1944.
Enniskillen Wesleyan Methodists:
Baptisms, 1823-1953; marriages, 1864-1906; circuit schedule book, 1866-80; quarterly meeting minutes, 1877-93.
Baptisms, 1841-1954; circuit schedule books, 1865- 1909.
Where there are no marriages in the Methodist records it would be because they were still using the Church of Ireland, even though their baptism and routine church services were in the Methodist Meeting House. This reflected the great reluctance amongst many Methodists in Ireland to separate from the Church of Ireland.
Just in case the family was Presbyterian, there is also:
Enniskillen Presbyterian church:
Baptisms, 1819-35 and 1837-1986; marriages, 1819- 34 and 1838-45.
If you are unable to go to PRONI yourself, you could employ a researcher. Researchers in the PRONI area: http://sgni.net
So to summarise, if looking for Martin’s baptism, if he was Church of Ireland and baptised in St McCartan’s, the records for the 1840s are lost. However if he attended Tempo Church of Ireland or was a Methodist or Presbyterian then there appear to be records you can search, but they are not on-line. I am sorry you made no progress in your visit to Dublin in 2018 but as you’ll realise from this, if they exist at all, the records you need are in Belfast.
Nearly all pre 1901 census records in Ireland have been destroyed, and there are limited sources for searching for someone born in 1841 who had left Ireland around 1859.
Have you checked marriage and death records in Canada for his parents names? If you can find his parents names, and they died post 1864, I can search for deaths. Have you checked passenger records to see who he was travelling with etc?