Theoretical physicist Roger Penrose is a proponent of this view, but theoretical computer scientist Scott Aaronson presents a rebuttal of his points [1]. Another article claims that the distance between synapses is two orders of magnitude too big for quantum mechanical effects to be effective, which seems like a plausible rebuttal to me [2].
There's regular quantum mechanics which underlies all chemistry and that you can use to calculate molecular properties and then the woo woo kind which Penrose seems to propose as behind consciousness on the basis that both are a bit mysterious so maybe one causes the other.