Please see the following:
New York (NY) – You can e-file an extension through the Department of Taxation and Finance. You can also print an extension with TurboTax that you'll mail in. Start your NY return (if you haven't already done so) and look for the extension question. If you don't see it, step through your NY return again from the beginning. The extension must be filed by April 15 and gives you another 6 months (October 15). If you owe them taxes, submit payment by April 15 to avoid penalties. You can pay when you e-file your extension or pay through their Online Services site.