The Standard Model is a well-known theory in physics that scientists think is (mostly) correct. According to the Standard Model, protons (and neutrons) are made up of quarks. Also according to the Standard Model, quarks (and electrons) are elementary particles, so they cannot be broken down further. Compared to a lot of other major theories in science, the Standard Model is quite new: scientists only really settled on the current version in the 1970s. Plus, like any theory in science, there's no guarantee that it's always correct. Still, most scientists--including me--believe that the Standard Model is "correct enough" to be useful. So far, all of the experiments in the world make sense under the Standard Model, so we often talk about it like it is fact.

But the Standard Model just one theory. Many physicists are working on theories that go beyond the Standard Model, and many of these theories say that quarks can be broken down. For example, the most famous theory that goes beyond the Standard Model is string theory. According to string theory, all of the elementary particles, including quarks and electrons, are made up of small objects called strings. This theory is really complicated and physicists are always thinking of new ways to improve it. So far, most scientists have not accepted any of these Beyond Standard Model theories as "correct": they need to be tested and improved more.

So far as we know, quarks and electrons are not themselves made up of smaller particles. However, that doesn't really mean that even protons (and neutrons) can be broken down, because these particles are made of energy, and the amount of energy you would need to make the quarks free from each-other is greater than the amount of energy needed to create new quarks, so if you hit a proton hard enough to break the quarks loose, you would just get more quarks to make mesons and things (a meson contains two quarks).