Quantum mechanics as a theory was invented in order to explain why classical mechanics and classical electrodynamics did not work in certain situations: small scales, spectra of atoms , black body radiation. Classical mechanics and classical electrodynamics cannot explain or predict the behavior of atoms (their interactions) , neither that of molecules.
Thus the answer is proteins interact with quantum mechanics. It can be mathematically demonstrated that classical mechanics and electrodynamics emerges from the underlying quantum nature when dimensions are large.
Now this:
(like my hand in my glove).
Even this depends on quantum mechanics. Scattering experiments have shown that most of the space around atoms, so that means your glove also, is empty. If the quantum mechanical theory were not at work, we would all fall to the center of the earth drawn by gravity after some time, to join all the other matter there. It is the Pauli exclusion principle that does not allow this.