It is definitely a bad starter issue.
However If there is no power at the starter, the problem could be the ignition switch.
Reliablity ratings for 1999 infiniti
Occasional problems on this vehicle are failure of the Exhaust Gas Recirculation (EGR) Valve Passages and failure of the Ignition Coils. Failure of the EGR valve passages will cause the Check Engine Light to illuminate. Failure of the Ignition Coils can cause the engine to run rough or the check engine light to illuminate. The cost to repair the EGR valve passages is estimated at $48 for parts and $325 for labor. The cost to repair the Ignition Coils is estimated at $461.61 for parts and $65 for labor. All prices are estimates based on $65 per flat rate hour and do not include diagnostic time or any applicable sales tax.