Approximate counting problem capturing BQP
Emanuele: Unfortunately, we don't know of any black-box problem capturing BQP as simple as the one you mentioned capturing BPP.

Intuitively, this is because it's hard to talk about BQP without bringing in unitarity in one form or another.  The ability to sum both positive and negative numbers is what makes BQP more powerful than BPP, but then unitarity is what makes BQP less powerful than #P! :-)

Having said that, besides the Dawson et al. paper that Martin Schwarz linked to, you should definitely check out this and this by Janzing and Wocjan, which give "surprisingly classical-looking" promise problems that capture BQP.

Also, let S &sube; {0,1}n, and consider a Boolean function f:S&rarr;{0,1}.  Then I have a conjecture from years ago which says that Q(f), the bounded-error quantum query complexity of f, is polynomially related to the minimum degree of a real polynomial p:Rn&rarr;R such that

(i) p(x)&isin;[0,1] for all x&isin;{0,1}n, and

(ii) |p(x)-f(x)| &le; &epsilon; for all x&isin;S.

If this conjecture holds, then an "approximate counting problem capturing BQP" would simply be to approximate the value of a polylog(n)-degree polynomial p:Rn&rarr;R, at a specified point on the Boolean cube, given that p is bounded everywhere on the Boolean cube.  This might be about as close as one could get to an answer to your question.