A heraldic-products vendor, House of Names AKA Hall of Names (Swyrich Corp.), has been claiming for years that there is a coat of arms for Conlish, Quinlish, and related names (the same one for all of them). This claim seemed dubious, because no arms for any such name can be found in any reliable sources on Irish (or British) heraldry, and the features strongly suggested they were ecclesiastical not secular arms.
Thanks to the College of Heralds in London, the arms have been tracked down in detail, and they are those of the Church of Ireland Bishopric of Clonfert in Galway, and have nothing to do with any of families of these or related surnames.
Details are now in the “Heraldry” page.