Not really a QX4 specific question here.
So I know you're supposed to use a brake grease on these, but I've spent too much money already and want to use something else I've got already.
I've heard some people use anti-seize, so I figured I'd just go ahead and use that. Is something terrible going to happen if I do?

Not worth the risk using the wrong grease where it should never be used. AutoZone, O'Reilly, Carquest, and many others sell single use brake caliper pin grease packets for about $1.

mdmellott has good advice here , but to "break it down " - you got metal on metal threads that I guess can be hit with anti-seize , IF , you use sparingly , BUT the actual SLIDE part of the pins , which have like silicone rubber 'condom' over it , definitely , like mdmellott says ought to use caliper slide -specific grease . So it's how you phrased the question , and everybody gets that ya just hate stuck pins - they break too easy .But I see it as two separate places to lube , and the threaded end only , going into the brackets are very probly OK w/anti-seize. Yeah! If that's the situation with yours , that they are a very much separated areas to lube , that the anti-seize doesn't get into the slide part . It may sound moot , to just get the caliper grease ? But I like how well antisieze works on tight threads , which see heat , like - have both - use both .