The Online New York Naturalization Records Guide has two pages:
New York City Naturalization Records (this page) and New York State Naturalization Records (next page)
Online New York City Naturalization Records and Indexes
There are a variety of courts in New York City where naturalizations took place. You may have to check more than one index. Some indexes and records may not yet be online. Some of the listings here may also include counties ouside of NYC.
The New York City Boroughs of today are Manhattan (New York County), the Bronx (Bronx County), Brooklyn (Kings County), Queens (Queens County), and Staten Island (Richmond County).
New York City Naturalization Indexes and Records at Ancestry (these require payment):