A workaround for the error caused by some missing functions of wines gdiplus (GDI+): Setup a new WINEPREFIX a.k.a. wine-bottle:

1. Open a Terminal
2. export WINEPREFIX=/path/to/wineprefix
3. export WINEDLLOVERRIDES="winemenubuilder.exe=" (to prevent useless desktop-files etc.)
4. run wineboot -f -u to create a clean Wineprefix
5. Quark need some files from VisualC++ 9: winetricks vcrun2008. This gives you the files mfc90u.dll, msvcr90.dll and msvcm90.dll. Only to know whats happens behind your back.
6. Quark also needs the original GDI+: winetricks gdiplus. The related file is gdiplus.dll
7. run wineboot -f -u again. (facultative)
8. Download the installation file from Quarks webpage and start the installer.
9. Start Quark with wine /path/to/QuarkXPress Passport.exe

Due GDI+ is still under construction you will have to check every new snapshot, if the workaround with winetricks gdiplus is still necessary.

From a comment here in the AppDB:
Quark with Ubuntu
by Odin on Monday September 6th 2010, 14:01
To put pressure on the Quark team to make a version for Ubuntu, go and tell them the need for it in this thread:

QuarkXPress on Linux/Ubuntu?!

All Quark users on a *ix-platform are called for support this thread!