Quark does not offer hanging punctuation. It is possible to define a hanging indent style to accommodate initial quotes, but the right hand margin would have to be variable according to content.
At one point, I worked on an Applescript that attempted to adjust margins (left and right indents) and text box size to allow for various left and right hanging punctuations, but I soon realized that mixed type sizes would scuttle the whole attempt, and I never finished polishing it. It also required changing multi-column text boxes to linked single-column boxes to allow for the box expansion, when you have runaround issues if the boxes overlap.
One other work-around for the left margin is to put a space character before the character you need to hang (possibly a quote) and kern the character right out of the box. It will become "invisible" on your layout, but it will print correctly.
It would be great if Quark implemented this feature, but I can see it's not easy, and I don't absolutely require it for my clients at this point. If I did, I would probably add InDesign to my desktop.