Sounds interesting...
I tink biggest problem is the same dabase structure.. But I think you can try by chaning prefix. So one q2a site will have qa_ and other q2a_ . While both will use same user table from WordPress.
This is my logic and havn't done prectically yet but would love to do...
>>>EDIT<<<
Okay cool... I have tried and install two Q2A with Single WordPress and seems working fine... Just changed prefix as I mentioned above.. see database in below image
But do not directly do on your production website.. Rather do on your dev environment and once you fine than work on live..