Is there a way to download just the sound/audio part of QuickTime
movie files onto an ipod nano? The nano does not support video but I
am only interested in the sound of these files anyway

The way to do this is very simple. No matter if you are on a Mac or run Windows. 
All you have got to have is Quicktime Pro. If do not have Quicktime
Pro here is what you do! If you are a Mac user Quicktime comes
pre-installed on your computer so all you need is to upgrade it to
Quicktime Pro, which will reveal you special features. If you use
Windows and do not have Quicktime player on your computer yet just go
ahead and download it from Apple's website then comes upgrade which is
the same as on the Mac. All information about upgrade you can find on
Apple's website:

http://www.apple.com/quicktime/pro/

and then:

1.open that .mov (video file) in Quicktime
2. go to File -> Export
3. give your file name then choose where to save it, then in export
menu choose: Movie to mp4 and hit Options button.
4. In the Options dialog box click the Video tab and from Video format
menu choose none, then go to Audio tab and choose copy or passthrough
(or just leave it as it is, sometimes does not matter) after all that
hit OK button.
5. Then just hit save and you are done!

OR you can make a workaround and directly from a Export dialog box
choose export Sound to AIFF and save it! Ipod Nano supports this type
of file as well.
In both cases then just drag that file into your own itunes playlist
or library. Next time you plug your Ipod it will be updated without
any problems!