Hi Sean,
Without knowing the townland or parish where your Michael came from, makes it just a little bit challenging trying to locate him in our County.
I've done a little bit of digging around on the Canadian end and it does appear that you may have more family than you thought, so I shall leave you to follow up on that.
Michael Moore
Buried: St Marys, Perth County, Ontario, Canada
https://www.findagrave.com/memorial/165125272/michael-moore
Michael Moore d. 18 Apr 1881, age 67, a farmer, Oxford, Ontario, Canada
Religion Roman Catholic. Informant: Robert Moore his son East Nissouri
*Robert Moore*
Hannorah O'Connor Moore
Buried: St Marys, Perth County, Ontario, Canada
https://www.findagrave.com/memorial/165125276/hannorah-moore
Hannora Moore d. 13 Feb 1892, Oxford, Ontario, Canada.
Religion: Methodist. Informant: Michael Moore.
The baptismal register for St Joseph, Stratford, Perth, Ontario, is available on Family Search which is a free website:
https://www.familysearch.org/en/
Please note the names of the sponsors of your ancestors, as the parents Michael and Ann would most likely ask close relatives, possibly their siblings to be the God-parents of their children.
William Moor baptised 3 Mar 1850
St. Joseph, Stratford, Perth, Ontario
Parents Michael Moor and Anne O'Connor
Sponsors John McLarkey and Johanna O'Connor
https://familysearch.org/ark:/61903/3:1:33S7-9YWH-LQ6
Note: William's youngest son Thomas Joseph Moore passed away in Detroit, Michigan on 1 Mar 1939, you will find this record on Family Search.
James Moore baptised 4 Sep 1853
St. Joseph, Stratford, Perth, Ontario
Parents Michael Moore and Ann Connor
Sponsors Michael Murphy and Margaret Patent (sic)
https://www.familysearch.org/ark:/61903/3:1:33SQ-GYWH-2ND?i=55&cc=1927566
Mary Ann Moore baptised 11 Oct 1857
St Joseph, Stratford, Perth, Ontario
Parents Michael Moore and Anna O'Connor
Sponsors Thomas Moore and Ellen O'Connor
https://www.familysearch.org/ark:/61903/3:1:33S7-9YWH-2BV?i=42&wc=M6VT-X...
Jeremiah Moore baptised 29 May 1859
St Joseph, Stratford, Perth, Ontario
Parents Michael Moore and Anna Connor
Sponsors Thomas Quirk and Marie Quirk
https://www.familysearch.org/ark:/61903/3:1:33S7-9YWH-GK4?i=65&wc=M6VT-X...
Johanna Moore baptised 22 Jul 1860
St. Joseph, Stratford, Perth, Ontario
Parents Michael Moore and Anna Connor
Sponsors James Moore and Mary Coffey
https://www.familysearch.org/ark:/61903/3:1:33SQ-GYWH-G7X?i=80&wc=M6VT-X...
Michael Moore baptised 2 Nov 1862
St. Joseph, Stratford, Perth, Ontario
Parents Michael Moore and Ann Connor
Sponsors Lawrence Murphy and Bridget ?
https://www.familysearch.org/ark:/61903/3:1:33S7-9YWH-2RD?i=72&cc=1927566
*James Moore and Thomas Moore*
Archives of Ontario:
http://www.archives.gov.on.ca/en/tracing/index.aspx
Land Records:
https://www.bac-lac.gc.ca/eng/discover/genealogy/places/Pages/ontario.as...
Wills:
https://www.bac-lac.gc.ca/eng/discover/genealogy/places/Pages/ontario.as...
The are 102 Moore's listed in the Tithe Applotment Books for County Monaghan alone. The Tithe Applotment Books were compiled between 1823 and 1838 as a survey of land in each civil parish to determine the payment of tithes (a religious tax).
Tithe Applotment Books:
http://titheapplotmentbooks.nationalarchives.ie/search/tab/results.jsp?s...
It would be "pure speculation" to suggest that this Michael is your relative, however these two gentlemen were the only Michael Moore's recorded in County Monaghan at that time.
Michael Moore, Townland of Carrolina, Parish of Aughabog, Co. Monaghan
http://titheapplotmentbooks.nationalarchives.ie/reels/tab//004239502/004...
Michael Moore, Townland of Feagh, Parish of Aughabog, Co. Monaghan
http://titheapplotmentbooks.nationalarchives.ie/reels/tab//004239502/004...
Civil Parish of Aghabog, Co. Monaghan:
https://www.townlands.ie/monaghan/aghabog2/
Carolina Townland, Co. Monaghan:
https://www.townlands.ie/monaghan/dartree/aghabog/aghabog/carolina/
Feagh Townland, Co. Monaghan:
https://www.townlands.ie/monaghan/dartree/aghabog/aghabog/feagh/
Irish Catholic Parish Registers are available here:
https://registers.nli.ie/
With every good wish,
Maggie McKenna, Carrickmacross.