Its fixed! It turns out the Contact associated with the Quote is not linked to the Company (Sounds obvious now). I had to go into each Blank Company Name Quote, and then into the Company Account. From there in the Contact tab there is an option to link a contact, search for the name in the window, select and close.
Refreshing the Quote list again and the Company name appears.
What is odd still is that under that Contact tab in the Company Acc the contacts are already listed. I would think they're linked already if they appear here?