I research the Irish Family History Foundation's database. It contains 12,700,000 surviving rcds for abt 70% of Irish counties. Rcds are increasingly sparse B4 1840, rare B4 1800.
Note 1 - Without a KNOWN county, the chance of an ancestor's rcds NOT being in this database is AT LEAST 30%.
Note 2 - Only 1/2 of Co. Cork's rcds are in this database.
Note 3 - The RC church kept control over most parish rcds from Tipperary (north). These are ONLY available thru the Tipperary Family History Research center at a substantial fee (regardless of outcome).
For Tho(mas) Quigley B 1885 (exact), Co. Galway, with a father forenamed Hugh, I found 2 rcds: 1-church, 1-civil.
For Hugh Quigley M 1881 +/- 4 yrs, Co. Galway, I found 1 rcd in 1881, Williamstown CP.
For Cat(herine) Lyons M ditto, ditto. These two are likely for your ancestors. A view of the detailed rcds SHOULD show they married each other. If so, the view MAY reveal their parents' names. This info is useful in identifying earlier ancestors.
For Hugh Quigley B ????, Co. Galway, I found 0 rcds within a relevant range. Hugh may have been from another county.
For Cat(herine) Lyons B 1860 +/- 4 yrs, Co. Galway with the father's forename unknown, I found 3 rcds: 1858 Bullaun RC parish, 1858 Killeenadema RC, and 1864 Dunmore CP. Father's forenames were Patrick, Michael, and James respectively.
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.