Known problems in BQP \ NP?
If there was a problem known to be in $\text{BQP}$ but not $\text{NP}$, that would prove $\text{BQP} \not\subset \text{P}$. But $\text{BQP}$ vs $\text P$ is also still an open problem.

It is suspected that $\text{BQP} \not\subset \text{NP}$. In fact it's suspected that $\text{BQP} \not\subset \text{PH}$ via the "Recursive Fourier Sampling" problem. But proving separations between complexity classes is surprisingly hard. I don't think anyone's even managed to show that $\text{BQP} \not\subset$ $\text{LSPACE}$.