Welcome to NICO! The 03-04 G35s for the most part, do not have any serious issues. However, there are a few small issues to keep a look out for. The Air Conditioner / Stereo circuit board and Stereo head deck are pretty well known issues. They will eventually fail (search the boards and you'll see numerous posts). Next would be that the brakes were a bit dusty. Not sure if there was a fix for it though. The headlight lenses can haze over. Not really a big deal, you just need to polish them with a good plastic cleaner. Finally, the paint on the front bumper is very susceptible to chipping. I had to have my bumper repainted...but then it was better than new and more durable.
Hope this helps!
Oh, and I would get a tire shop to balance your tires and do an alignment for the wheel shake
