Yes, electrons do move from atom to atom in a conductor. In a conductor, such as a metal wire, the electrons in the outermost shell of the atoms are loosely bound to the nucleus and are free to move throughout the material. When a voltage is applied across the conductor, an electric field is created which causes the free electrons to move in a particular direction.
The flow of electrons in a conductor is due to the movement of these free electrons. When a voltage is applied across the conductor, the electric field exerts a force on the free electrons, causing them to drift in the direction of the electric field. However, the electrons do not move in a straight line, but instead collide with the atoms and other electrons in the material, causing them to lose energy and change direction. This results in a random, zigzagging path of electrons through the material, known as electron drift.
Despite the random motion of individual electrons, the overall flow of electrons in a conductor is in the direction of the electric field. This is because there are a large number of free electrons in the conductor, and their collective motion results in a net flow of charge in the direction of the electric field. This is the basis of the electrical current in a conductor.