I am guessing that XP's DOS doesn't support whatever the call is to read the key disk. I had some trouble with the game Lemmings that required the disk to be in the drive. I know the disk is ok because I can play it on the old 486 I have for DOS gaming, but it wouldn't work on my XP machine.

Funny that you should mention Qix, my son was just playing it on the Atari 800 a couple days ago. You might be able to find it on the net for another platform such as the Atari and play it though an emulator, though I am not sure if they ever released the license for it. Probably not, since a quick search shows it in the recently released (UK) Taito Legends 2 for the PS2/Xbox.

Most likely, it is protected in the manner very similarly to Arkanoid. Does not work on XP either -- XP does not support some low-level floppy controller calls. Win9x allows more direct control so it might work. It may still fail with some modern floppy controllers.
Atari 800 version may actually be better than PC one -- that's not uncommon for games of that period. My favorite version of Arkanoid was the one for MSX, I would not be surprised if Qix was good on that platform as well.

Note: I tried it with DosBox as well, and it wouldn't work, so possibly the floppy controller is also part of the problem. Any solutions, i do not have an Atari or any other system.

Just Google "qix dosbox" and check a few first hits.