I don't know of an example from BQP, but 3d protein folding is NP-complete and humans have outperformed computers at protein folding. Although I don't know how you'd even directly compare humans and computers. I suppose that's a big part of what this article is grappling with.

The interesting thing to me is not if humans can solve exactly the same problems as quantum computers with lots of qubits (Quantum computers are not the identical to all of quantum theory!) The interesting question to me is if humans can hold and pass around irreducibly probabilistic states without collapsing them.