if you have the missle lancher mouted on your ship you should see it on the weapons list (eraser missle etc.) just press the number next to it..
so if you have...
1. Justise MK 1
2. Justise MK 1
3. Eraser Missle

Press 3 and your missle will fire

you need aswell have to have some ammo for your launchers ^-^

Just to be clear, you need to match your launchers with ammo (ie. You need Javelin Launcher for Javelin Missiles). You might come across ammo that has a launcher not sold at your current location, so that might cause some confusion. Pain in the butt, really. It hardly makes sense how every launcher has it's own exclusive missile.

but once you have the launcher, every base has the ammo.

On top of that, the game highlights the ammo that you can use, and greys out the stuff you don't have a launcher for.

Reaper: $200,000,000.
Rockets: $100,000
The look on that pilot's face when you launch a full pod of screamers up his tailpipe: priceless

Also if its a Cruise disrupter thats a different key

Fire Cruise Disruptor = Shift + E

You might want to check your keyboard controls just to make sure that Q is still missles.

Personally, I change the weapon keys depending on what kind of ship I'm flying. For instance, freighters and light fighters don't have torpedoes, so that key becomes mines... or missiles, if I'm using them.