It's not entirely infeasible that neurons could harness quantum effects. Not across the neurons as a whole, but via some sort of microstructures or chemical processes [0]. It seems likely that birds harness quantum effects to measure magnetic fields [1].

0: https://www.sciencealert.com/quantum-entanglement-in-neurons... 1: https://www.scientificamerican.com/article/how-migrating-bir...