The term Irish refers to anything originating from the nation-state of Ireland, including its people, as well as the Irish language.
The shilling was an Irish coin during the 19th century, and remained in use until 1971. An important festivity was Saint Patrick's Day. (VOY: "Fair Haven")
Jean-Luc Picard carried a copy of the Irish novel Ulysses with him to Risa. (TNG: "Captain's Holiday")
The Bringloidi were colonists of Irish descent. (TNG: "Up The Long Ladder")
In 2365, a hologram representing a comic mentioned an Irish person in one of his jokes. (TNG: "The Outrageous Okona")
An Irish couple was among the survivors of the "37's", found by the crew of the Voyager in 2371. (VOY: "The 37's")