Indeed, protons are quantum particles, they are fermions, and must also satisfy the Pauli exclusion principle (fun fact: that's why we exist). The interesting thing is that they also distribute in energy levels, that's what we call the Nuclear Shell Model.
The forces are different, also the interaction, so the levels look different, but they have levels too.
I understand that this may be counterintuitive. We tend to imagine electrons as if they were in orbits (the way Bohr described them), but that is not what's going on. Electrons distribute in levels that we call orbitals, and they are levels in energy. In the space these orbitals are clouds of probability, so in reality you just know the probability of having an electron in a certain position. For this reason, we can no longer think of a planetary system like picture.