Union League was the name of a building on Fifth Avenue, in New York City. This building, described as a "...gray old beaux...", along with the Metropolitan and the New York Yacht, housed the Union League Club of New York. This was an exclusive club for wealthy New Yorkers in the early 20th century.
In The Big Good-Bye, this was one of the attractions on this avenue that drew the attention of the master criminal Silent Forrester as he was being chauffeured to the hotel Plaza. (TNG: "The Big Goodbye")