Many people claim that the electron does not move. Without realizing it, they make a mistake and assume that you are referring to a state with definite energy. These states are not dependent on time and, therefore, have no motion in a classical sense.
Quantum mechanics elementary problems are often focused on energy states. This can lead to students believing that electrons are always in well-defined energy states.
However, electrons do not have to be in this state. They can move around the nucleus if they are in a superposition or energy state. They are most easily observed in high-excited states. They can then appear to be moving around a nucleus much like a planet orbiting the sun.
These orbits are usually of low velocity. According to the incorrect but informative Bohr model we have circular orbits. The electron in the lowest orbit moves at 1/137 speed of light. This model is incorrect. You need at least two orbitals with different energies to get electron motion. This is easy in quantum physics. For example, if you place the electron in a superposition between the lowest energy state of the electron and the first excited states (s and orbitals), the wavefunction will oscillate from one side to the other, which can be taken as the electron moving. Because these motions involve less negative energy states than others, the velocities that you get will always be lower than 1/137c.