Hi Dutch Dave.
OK, this could be a long one or a short one depending on how accustomed you are to Colour Management in Quark.
Basically, what you need to do is get Quark to put it's colours out in sRGB colour, so that they look good on a generic monitor.
First step is to set up some colour setups.
1. With no documents open go to Edit>Color Setups>Source... In this box, click 'New'
2. Under the RGB tab, you will see that there are two pairs of Menus, one for images and one for solid colours. In each, change the profile to sRGB (or sRGB 61966-2.1) this means is that any colours you create in Quark will be created in the sRGB colorspace, images that you import will be assumed to have been created in the sRGB space unless they have a different profile (such as Adobe RGB) embedded into them in which case they will be converted using the embedded profile. BE SURE TO CHECK THE MANAGE RGB SOURCES TO RGB DESTINATIONS box.
3. Under the CMYK and other tabs, you can leave it as it is unless you know exactly what CMYK profile you are using for prepress work. Name this Source setup as sRGB Source setup. Click OK/Done to save it.
4. Now go to Edit>Color Setups>Output (this output setup will be useful if you want to print anything on a consumer inkjet or make an RGB PDF that will be viewed on a generic monitor such as a PC)
5.Click New. Select Composite under Mode; RGB under Model and select the sRGB (or sRGB 61966-2.1) profile. Name this setup as sRGB output. Click OK and save.
6. Now, go to QuarkXPress>Preferences...
7. Scroll right to the bottom of the left hand list in the preferences dialog and select Colour Manager under Default Web Layout. In the Source options sections change the Source setup to your newly created sRGB Source Setup and tick Enable Access to Picture Profiles.
Under HTML export select the sRGB (or sRGB 61966-2.1) profile. Click OK
8. There's a Colour Manager section under the Default Interactive Layout list on the left. Set exactly the same options in here too for Source setup and SWF export.
Doing all of this without a document open will make these changes appear on any new document you create. To make these changes apply to existing documents, you'll have to change the preferences on those too.
Hopefully now when you export or preview, you should get much better colours. There may be other considerations, like how your monitor is profiled for example that may affect how this all looks, but this will make a good start for you.
If you need any more info, just shout.