Apple Inc. did sue NYC, however it was not about "Big Apple" nickname, but the logo that went along with it — a stylized apple.
Apple filed a federal challenge to New York's trademark application
for a new "Big Apple" logo, saying it's too similar to the stylized
emblem found on iPhones, iPods and iMac computers.
Cupertino, Calif.-based Apple said the symbol for New York's "GreeNYC"
initiative promoting energy efficiency and recycling is confusingly
similar to the logo used by the electronics maker since 1977.
NYC & Company Inc., the city's nonprofit tourism and marketing office,
filed the trademark application in May, playing off of New York's "Big
New York already has begun using the logo, which morphs the symbol for
infinity (similar to a figure 8 on its side) with the outline of an
apple, a stem and a single leaf.
Source: "Apple sues New York over logo", LA Times
At the same time Apple filled opposition in USPTO,
which sided with NYC, dismissing the opposition.
The story was also covered on Wired.com.