As some cool Celtic musicians once said, “Everyone is 1/365th Irish.” You know, because of St. Patrick’s Day!
However, if you have some real Emerald Isle blood in you (like 12% of Americans), and your ancestors landed in Utah (and maybe stayed awhile), there is a chance the Utah State Archives Research Center has some records about them.
Naturalization records were created as immigrants started and completed the process to become American citizens. The process was a tad harder to track prior to 1906 when the predecessor to the Bureau of Citizenship and Immigration Services standardized the forms and procedures. For example, before 1906 an individual could become naturalized in any court in the land, regardless of where they lived (or passing through). If you can untangle the likely courts and jurisdictions to look through, there may be a variety of aids like indexes (either online or more likely in the actual record books) and inventories of holdings.