The availability of a quiz for students to take is controlled by the settings of a Start Date and an End Date on the "Restrictions" tabbed page of a quiz.
The Start Date specifies the earliest date and time that a student will be able to begin the quiz. To set a Start Date, check the box by "Has Start Date" and set a date and time.
The End Date specifies the latest date and time that a student will be able to begin the quiz. To set a End Date check the box by "Has End Date" and set a date and time.
A student who begins a quiz before the specified End Date will be always be allowed the full amount of time specified by the time limit. For example, suppose a quiz with a one-hour time limit has a Start Date of 4:00PM and an End Date of 10:00PM on a particular date. If a student begins the quiz at 9:23PM on that date, the student will have until 10:23PM to complete the quiz, even though the End Date specifies a time of 10:00PM.