The practice is common in certain sectors of the US education market. It appears less common in other parts of the world where teachers make other demands on Moodle. This is not supported as a standard feature.
In recent versions you should be able to set an expected completion date using activity completion that is before the due date if you want to add a date to the calendar for the quiz before the quiz is closed. You would then need to penalize the students manually for submitting after the expected date.