Answer:
No. New York City is not in any one county. It is the opposite, in fact: the counties are in New York City.
New York City is comprised of five boroughs: Manhattan, Brooklyn, Queens, The Bronx and Staten Island. Each borough is also a county: Manhattan is New York County, Brooklyn is Kings County, Staten Island is Richmond County, and Queens and The Bronx have the same names (Queens County and The Bronx County).