thing:Sorry, perhaps I didn't make my suggestion clear. Import the Illustrator file - it'll bring it's version of PMS 871 with it. Then use that colour instead of making a Quark version, or delete the Quark version and when it asks what colour to use as a replacement select the illustrator imported PMS.
Alternatively, if you don't need the PMS colour to be a spot open both and make the CMYK mix match exactly. Job done.
Greg
This is one of my pet hates. With quark7, unlike every other version of quark THIS DOESN'T WORK, AND ITS INTENDED NOT TO WORK BY QUARK
Change the name of your PANTONE spot colour in Illustrator to something other than PANTONE then follow things instuctions and you'll get the cmyk split you want.
If an imported Spot Colour has the same name as one in a quark7 PANTONE library then the elements coloured up in quark use the quark library numbers and Lab as an alternative composite colour space, so if the numbers dont match between illy and quark you get two different cmyk splits when you output, you need either to print as a spot colour, or print as a spot colour and use Pitstop or something to remap the "two" spots (the same name but different alternative colour spaces) back to a common cmyk break.
At the end of this excersize you can actually be grateful that Quark can't put transparency in a pdf. 
try this at home:
In Illustrator create a spot color swatch make it a red Color and Call it PANTONE 539 C (which is a dark blue), create an object and color it up then save it as an EPS.
Create a brand New quark document and place the EPS, this will bring into the color palette a Dark Blue swatch called PANTONE 539 C.
Next create a box in quark and color that with PANTONE 539 C. so on screen you now have a red item and a blue item.
Export to PDF as spot color.
Look at the PDF both elements appear Red, (Ah Ha, this is what I want - I think) but convert that pdf to cmyk and one element is dark Blue one is Red.
Now I know this is an extreme example, but follow the logic where the color shift is slight and the PDF is put to a RIP which converts all spots to CMYK and you have an unhappy corporate.
... 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