Welcome to NICO.
First thing I would try is to lock and unlock the doors from the outside with the key. The old Q's had an issue with the switches getting worn out and it woudl unlock the door but not disarm the security system. most the time physically unlocking the door with the key would help this.
Next thing I would check is the starter relay. It should be a big blue relay located behind the passenger side headlight (or thereabouts).. You could try jumping the contacts with a paper clip or something on the starter relay connector, and see if the starter goes. If it does, that means you've likely got an issue with the starter relay or with the security system. If it does not, then that means you've probably got a bad starter.
The starter relay is a common failure item on the Q. Sometimes you can bring the starter relays back to life by unplugging them and hitting them against a table top or something. Corrosion can tend to build up inside and foul the contacts, and sometimes you loosen it up enough with a little jarring action..
