Since "Full name" is an optional field, then obviously there's no read-made toggle option for you to click.
You have to override function post_meta_who, (or you can develop your own function to display whatever you want).
"Full Name" is a field in user profile, so it must be stored in qa_userprofile table. You may also double check as I can see there's another similar table named qa_userfields. I believe it depends on how you set up your site.
So you do a MySQL query to such a table, and then replace the default $post['who']['data'] in post_meta_who function. You have to prepare for the case that your users don't have a Full Name, or they use this field to insert harmful scripts or SQL injections.
In my own Q2A site, I create a new unique function that displays the post author information with the similar look like in Youtube or Reddit.