No - You have to file a separate extension request for New York. You can prepare your NY extension through TurboTax, but you will have
to mail it in.
You can prepare and submit an extension online on
the NY website here: NYS online service.
The
extension must be filed by April 15 and gives you another 6 months (October
15). If you owe NY taxes, submit payment by April
15 to avoid penalties. You can pay when you e-file your
extension or pay through their Online
Payments site.