There are lots of opportunities to convert 100%k into 4colour black between an ai->qxp->jaws/distiller->pdf.
Quark8 also does some unexpected stuff too. More so when Exporting via Jaws rather than Distilling.
I suggest you turn off the tab to embed the color profile when you save your illustrator file.
You don't say whether you are saving as .ai or .eps or .pdf but this could influence the pdf out of quark. My Preference (for quark) would be .eps, but if you get no joy then try one of the others.
In Quark's Preferences>Color Manager turn off the checkbox that says Color Manage Vector EPS/PDF
If you have enabled Enable Access to Picture Profiles you may have to turn off anything in the Window>Profile Information (select the placed pic) that overrides the defaults.
Now if you choose Composite cmyk or Composite cmyk & spot or As Is you should get the same result unless your Illustrator black is RGB or Spot. But there is further chance to convert to cmyk via profiles in Distiller Settings so check that they are to Leave Color Unchanged and ignore profiles.
As Is is my favored output mode for PDFs from quark8 but its not a completely Color Management free workflow, I'm still trying to sort out what on Earth is happening to placed Spot Colors. (I'm in Prepress for a litho Printer so hope this puts your mind at ease)
It's also possible to set up As Is to embed ICC profiles but hopefully you've not gone down that route, since it is a whole new can of worms converting ICC to Postscript Color Management.
Finally if you are using Distiller (or Jaws) to embed profiles or tagging profiles or using PDFX3 or PDFX4 where there is a referenced intent, then What you use in Acrobat's Separation Preview must be matched to the profile or intent of the object to give the actual %numbers rather than the converted to Simulation Profile %numbers.
... there was the puzzle of why the sun came out during the day, instead of at night when the light would come in useful.
Terry Pratchet