Hi there and welcome
in OS X Lion, Apple has removed the capability of running older, PowerPC-only code. Before Lion, PowerPC code ran in an emulation layer called Rosetta. But in Lion, programs that require PowerPC code will simply not run.
This is why it will not run
QuarkXPress will need to be upgraded and you can purchase it here
If you would like any additional information or assistance, please do not hesitate to let me know.
Else if you can take a second to rate my service by clicking one of the stars at the top of the screen then submit, that would be greatly appreciated.