If your Check Engine Light is one, that I would highly suspect a failing throttle position sensor, this requires replacement of the throttle assembly.
However I really think you are more likely looking at either a faulty Mass Airflow Sensor or a loose connection somewhere in the air intake after the mass airflow sensor.

That would be the first thing I would check and if you have and stored codes if you could provide them we can help you out a little more.

Nico has a section dedicated to the G35 please feel free to check it out.
http://forums.g35club.org/

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http://forums.g35club.org/zeroforum?id=262

You may want to think about calling your local dealers and see what the going rate for a full check over.

Let us know if you need any help,

When you get a chance go check out the G35club segment. They have all kinds of interesting things you can do to your car.

Enjoy your new G.