Will a bad alternator make the computer think that other parts are bad like the starter or the brake lights to stay on or symptoms of a bad starter?

No, but a bad alternator can deplete your battery and cause starting problems. It won't make your brake lights stay on, that's most likely a problem with the pedal switch. If your Q has the same setup as most old Nissans, there's a rubber bumper on the pedal frame that acts as a switch-stopper. If it disintegrates or falls out then the switch won't depress and your brake lights will stay on all the time.