It sounds like the battery in your key fob is low or dead. In my G37 there was a port that you can plug the key fob in to start the car when the battery is dead, not sure if the GT-R is the same but I believe so.
Not to offend you but is your foot on the brake? Is your cars battery dead? Is it fully in park? What year is your car, 2011? I have heard about the 09's having steering colum lock problems which may cause a problem when starting but I'm not 100% certain on this!