Yes, protons move within the nucleus of an atom. However, their motion is different from the motion of particles in macroscopic objects. The behavior of particles at the atomic and subatomic level is described by quantum mechanics.
According to quantum mechanics, particles such as protons do not have well-defined positions and velocities like macroscopic objects. Instead, they exhibit wave-like properties and are described by wave functions. The wave function of a proton within the nucleus represents the probability distribution of finding the proton at different locations within the nucleus.
While protons do not move in the sense of classical particles, they can undergo quantum tunneling, which allows them to occasionally cross potential barriers within the nucleus. Additionally, protons can interact with other particles within the nucleus through the strong nuclear force, which binds the nucleus together.
It's important to note that the description provided here is a simplified explanation of the behavior of protons within a nucleus based on our current understanding of quantum mechanics. The actual behavior of particles at the subatomic level can be quite complex and is still an active area of research in physics.