Quantum computers are not known to be able to solve NP-complete problems in polynomial time. (Assuming for the sake of argument that we can actually build them.)

There are problems -- factorization, as others have already said, is the commonest example -- that (1) are not known to be solvable classically in a reasonable amount of time and (2) are known to be solvable by a quantum computer in a reasonable amount of time. But they aren't NP-complete problems.