If you allow for variations in the translation and spelling of names and nicknames and for people who neither remembered nor knew nor cared exactly when they were born, you will find several baptisms for this family in the registers of Tynagh parish at
https://registers.nli.ie/parishes/0339
The following are well enough indexed at ancestry.com to come up in a search with judicious use of wildcards (father ja*s quin*, mother moon*y):
View Record John Quin Baptism 28 Apr 1824 Tynagh, Galway James,
Biddy
View Record Michael Quin Baptism 24 Jun 1827 Tynagh, Galway James,
Bridget
View Record Kitty Quin Baptism 31 Jul 1829 Tynagh, Galway James,
Biddy
View Record Mary Quin Baptism 31 Jul 1829 Tynagh, Galway James,
Biddy
View Record Patrick Quin Baptism 19 Mar 1832 Tynagh, Galway James,
Bridget
View Record Marianne Quin Baptism 10 Oct 1834 Tynagh, Galway James,
Bridget
You may find others not so well indexed.