Ok,, very interesting.. you are getting exactly the same error as the TC gets.. which is interesting.. basically the OS on the Mac has lost track of where the disk is..
There is a long thread about the error -6584.. but it doesn't seem to have a clear resolution.
I will look up the thread.. this forum has the worst search... ever.
Did you mount the sparse bundle and verfiy it??
See A5 in pondini troubleshooting.
http://pondini.org/TM/Troubleshooting.html
Is the Mountain Lion a clean install?
Can you ping the NAS by name when the disk is lost?
Do you enable the server with username and password in finder? I am not sure of the specific instructions for setting up the QNAP with TM.
I would encourage you to use another backup software.. CCC, SuperDuper, Chonosync are the typically quoted ones. TM gives more trouble than its worth sometimes.
You can get a check of the sparsebundle by using the option click on the TM icon.
Anyway this is the thread.
https://discussions.apple.com/thread/3123920?tstart=30