Answer:
No, Brooklyn is one of the five boroughs of New York City: Manhattan, Brooklyn, Queens, The Bronx and Staten Island.
However, prior to the year 1898, Brooklyn was an independent city (not part of New York City). Before 1898, New York City consisted mostly of just Manhattan. In 1898, the five boroughs consolidated to form modern-day New York City.
Brooklyn was the only one of the four that was its own city before becoming part of New York City. The other three (Queens, Staten Island and the Bronx) were counties (or parts of counties, in the case of the Bronx) consisting of small towns and farmland.