You can say,
I'm Northern Irish
저는 북아일랜드 사람이에요.
or
I'm from Northern Ireland
저는 북아일랜드에서 왔어요.
I think when you introduce yourself as Northern Irish to Koreans, you are asked to introduce the historical background of Northern Ireland.
Usually Korean people don't know why you are a citizen of UK, even though you live in Northern Ireland.
I hope this answer would be helpful for you.