You cannot delete messages, but you can hide them so that they no longer appear in your Quora inbox.
To hide a message thread between you and another Quora user, visit your Messages page and click on the thread you want to remove. Then, click on the "Hide Messages" link in the upper right:
Note: Once you’ve hidden a message thread, you cannot un-hide it. The only way you’ll see the thread again is if you or the other user sends a new message.