If the Alt plug is a four pin connector with probably just one wire in it---Then it doesn't need to go to the PCM. If it is a two wire connector with two wires in it---Then it is set up for a PCM PWM alternator and does need to go to the PCM.
The single green wire in a blue connector sounds like the temp sender wire for your gauge. There is also a two wire temp sensor that inputs to the PCM.

Jeff