I grew up in Queens and now live on Long Island. I hunt Columbia County which is now a 2 &1/2 hour drive, but I wouldn't trade it for the world. you can get an Access Permit, then the hunting tag from the DEP to hunt the Watershed Properties in the Southern Zone. There very close to NYC, as they supply the water for NYC. And there's the land as others have said as well, Westchester, Putnam, Dutchess, Rockland, Sullivan Counties are all close, then Connecticut as well.
Long Island is tough though as there are very few landowners who grant permission and the public lands are overflowing sometimes with hunters.
Hope this helps, and thanks for serving our country and protecting our freedoms.