One problem related to battery dead has been reported for the 2004 Infiniti G35. The most recently reported issues are listed below.
On March 11 my 2004 infinity G35x experienced a complete alarming system failure. The power steering failed, the brakes failed, in fact every component not running on the battery failed. The dash warning lights looked like christmas lights. I was driving at the time. Fortunately I was on a side street, and not a highway. The dealer diagnosed a p-0340 code and replaced bank 1, cam sensor. This vehicle has just over 41,000 miles and is out of the warranty period. I noticed on-line that this part was part of a voluntary recall in model years 2003 and 2005, but not my year, but also that the failures reported were at approximately the same mileage as my vehicle. This could have resulted in a catastrophic accident had I been on a street with more traffic as I had only brute force control over the brakes and steering. My impression at the dealer was that they were very familiar with the display of warnings on my dash panel. The diagnosis took very little time, the repair was costly, the experience was extremely memorable.