To backup all the files you need to go to phpadmin and then follow this...
go to myphpadmin > select q2a > export (maybe at top navigation) > SQL (from drop down) > go.
Save the file and you are good to go.
You have made a backup so next thing would be to move your site to another directory, for that:
Make a new directory folder (assuming it to be qtoa as you want) then upload Q2A files to that directory and in qa-config file set the same database name as you previous one. When you open the site, go to admin > general and there change the preferred url to you new url (like www.example.com/qtoa).
Now you are all set to go.
You can delete the old Q2A files from you main domain as those are not needed anymore. But just don't touch your database.
I hope this helps.