Quantum electrodynamics is used to explain how the Electromagnetic and Weak Nuclear Forces work.
Quantum Chromodynamics describes how the Strong Nuclear Force works.
It has been theorized that the quarks, which make up protons and neutrons, are themselves comprised of elementary-er particles called preons:
Can preons be described by a quantum dynamic theory?
Quantum electrodynamics, quantum chromodynamics and quantum flavordynamics (more commonly known as the electroweak theory) are examples of quantum field theories (the -dynamics suffix just an accident of history).
No-one’s developed a quantum field theory based on the preon model that is in agreement with experiment as far as I know (and I’m surer if they did it would be massive news). My gues would be is that obviously searching for a QFT would be the starting point, but it might need something beyond that. Currently we have the Standard Model which is not a quantum field theory in itself, rather a framework for connecting two quantum field theories.
We’d need a lot more experimental data before we could even start constructing such a theory, and it wouldn’t necessarily be necessary, anyway. The strong force can, for many purposes, be described directly as a force between baryons, mediated by mesons, rather than as an emergent property of the color force between quarks, mediated by gluons. Likewise, if there’s some underlying layer below quarks, you could still probably get by with the quark-level theory. And the quark level theory is presumably simpler than the underlying one, and we still haven’t gotten that to work completely, either, so that would just make the underlying one all the harder to find.