I searched the Irish Family History Foundation's database. It contains 12,000,000 surviving rcds for abt 65% of Irish counties. Rcds are increasingly sparse B4 1840, rare B4 1800.
For Martin Concannon M ???? in Co. Mayo, the earliest M rcd I found was for 1822 in Kilcolman parish
For Bridget Noon(e) M ???? ditto, I found a M rcd in 1822 in Kilcolman parish.
The above are matches. It is likely that these M rcds are for your ancestors. A view of the detailed rcd will show whether the marriage was to each other. If so, then the rcds SHOULD list their parents' names, etc.
For a Patrick & Bridget M in 1860, I found M rcds in 1860 in Co. Mayo. Will leave it to you to determine if the M rcds were created in the same parish. If so, then these are likely for your ancestors.
For a James & Mary M in 1840 (this date seems too early), I found 0 matching M rcds. A more likely date for this M is 1862.
Possible siblings using a father's forename of Martin are: Mary (1842 & 1844), Ann(e) 1854, Bridget 1837, Catherine 1836, James 1840, Michael 1854, Martin 1856, Patrick 1846, and William 1839.
Google the Foundation's name to lead you to this database. Register, opt to be notified as rcds are added and browse. Pls post your search results to my genforum msg.