This is funny–I just published (this week) a museum catalogue in which I spelled it both ways in single paragraph. I quoted F. Scott Fitzgerald spelling it “Queensboro” but in my own text I spelled it with the “ough” and my editor asked me to be more consistent. I said I was being consistent–Scott spells it his way, and I spell it mine.
Sorry! I am from the Bronx and genetically incapable of finding my way around Queens, even thirty feet from the East River. 
However, I did find the Socrates Sculpture Park and MOMAQNS last year, so maybe there’s hope.
According to the USGS and the Board of Geographic Names, it’s the Queensboro Bridge
http://geonames.usgs.gov/pls/gnis/web_query.place_code_GetDetail?tab=Y&id=962088