A Japanese quartz movement is a mechanism for keeping time based upon the regular vibration of tiny section of quartz crystal. The "Japanese" part of the name refers to where the mechanism was manufactured, not where the crystal originated from.
All watches and watches need a regularly moving object, called an oscillator, to ensure that the time they are keeping does not become erratic. A familiar example is the swinging pendulum of a grandfather clock. Clockwork-based watches use physical levers, springs and cogs working together to produce regular and precise rotation of the hands around the face of the watch.
Quartz watches do not need the elaborate physical parts of clockwork watches. Their oscillator is not a pendulum or spring, but the quartz crystal itself. The crystal vibrates in response to a small electric charge constantly applied to it. Scientists call the vibration the piezoelectric effect, and the amount of vibration can be very precisely known, allowing for extremely accurate time-keeping.
The vibrations of the crystal send pulses to a circuit which count them and make the hands of the watch turn. Because no mechanical parts are ensuring the watch keeps time, watches with quartz mechanisms do not need to be wound.