It’s not possible inside Q2A, because messages (private and public wall posts) are always tied to a userid in the database.
It would be possible to do something similar yourself with a page plugin. The page would accept a parameter of the username/id (e.g. example.com/send-pm?user=1234) then the page would have a form like the feedback page. That would at least send the PM, but you’d need to create a separate database table if you wanted to store messages. And to be honest that kind of system is ripe for spam.