Is 2007 the first year that Infiniti made the ''push start'' ignition on the G35

You're correct. The 2003-2006 G35 is all have a more traditional ignition switch where you insert a key. Infiniti is 2007 models with push start began with the M35 and M 45 models.

The above answer is correct with the exception of the M35 bit; The new M35 was introduced in 2005 as a 2006 model, and it featured the Infiniti 'smart key' with push button start.