Baptism records prior to 1864 would be Parish records as civil registration didn't begin until 1864.
Galway East and West belong to an Irish History Foundation, it is a not for profit but fee paying website and about the only way to find records on line before 1864.
After 1864 try: www.familysearch.org
Martin Cunnealy born 25 Jan 1868, Williamstown, Galway
Fa: John Cunnealy Mo: Margaret Mcdermott
Templetogher is a Parish in the Poor Law
Union of Glennamaddy. Did not find Carrownearry but when searching for McDermott's there were McDermott's in the Townland of Carrownderry, which is probably close enough to be the right place.
Griffith Valuation of 1856:
No Cunneally/Conneally listed in the Parish of Templetogher
but there are three in Galway:
William Cunnally, Parish Boyounagh
James Cunneally, Parish Moylough
James Cunnealy, Parish Moylough (possibly same person but
surname spelled differently, very common in Ireland.
The McDermott's were in Griffith Valuations also with
215 in Galway, 60 in the Parish of Templetogher.
A marriage probably occurred in the bride's parish.
There is a William McDermott living in the Parish of Templetogher and the townland of Carrownderry, he is listed 3 times there as he OWNS land and 1 in the Townland of
William McDermott and Glasgow Connolly own land and
properties 1-19 and rents these out to 17 persons a total of 171 acres 3 roods and 36 perches
#16 & 17 a plantation 2.1.20 and land 1.0.4 are owned by
The Townland of Carrownderry
#1 William McDermott OWNS/IN FEE bog 92.1.11
#15 Plantation 1.0.15 & cottiers house and land 0.0.12
rents out remainder to 19 people, total property 272.2.32
The Townland of Gortdrishagh
#1-4 William McDermott & Glasgow Connolly owns a plantation and rents out houses and land totalling 54.3.5
There are other William McDermott's IN FEE/OWNED lands in Townlands Castletogher, Cornanaff, Liskea and Monasterowen.
A wealthy man.You can see these at www.askaboutireland.ie
He is not listed as a Gentleman or Nobility in the Directories of Ireland. You can see examples of early ones 1824 & 1846 on www.failteromhat.com
1901/1911 census: www.census.nationalarchives.ie
Tithe Applotments: 1820-1839 www.genealogy.nationalarchives.ie