The closest direct equivalent of "nerve" in this sense is probably {butlh} ("gall; dirt under one's fingernails"). However, this is generally seen as quite a positive trait to have; something to say if you're a bit impressed that he's trying to charge you for the damages.
You could say something like {quv Hutlh} or {nur Hutlh} to describe somebody who acts shamelessly, or {nguq} for somebody with an overinflated ego, or {tlhIv} if they're challenging somebody who outranks them.
For the example with the damages, I'd say {quv/nur Hutlh} makes the most sense, because the person seems to be out to make material gains without regard for what people will think of them; they'd rather be rich and reviled than poor and admired.
If the person is trying to claim damages in order to give off an appearance of innocence, {Qaq} might be appropriate.
If speaking directly to the person, {bItaHrup'a'?} may get some of the idea across.
http://klingonska.org/canon/2003-09-holqed-12-3.txt
//loghaD