Long Island City is in the NYC borough of Queens. "Long Island" is usually understood to mean Nassau and Suffolk Counties -- the two counties east of Queens that are not part of New York City.
However, it should be noted that the New York City boroughs of Brooklyn and Queens are located geographically on the large island known as Long Island.
Queens is the gateway to Long Island but it is not Long Island, it is Queens. Queens is a borough of NYC which Long Island is not.