Yes, you need to file a NY State Extension separately.
You can e-file an extension through the Department of Taxation and Finance, or you can print an extension within TurboTax that you will need to mail in. In TurboTax, start your NY return by clicking "State" on the left side of your screen 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) to file. If you owe NY taxes, be sure to submit your payment by April 15 to avoid penalties. You can pay when you e-file your extension or pay through their Online Services site.