Currently there is no way to do this.
It would be possibel to get the desired feature without changing core Moodle code, but creating a new Quiz acccess rule plugin https://docs.moodle.org/dev/Quiz_access_rules
It would be possibel to get the desired feature without changing core Moodle code, but creating a new Quiz acccess rule plugin https://docs.moodle.org/dev/Quiz_access_rules
If, if you could find an elegant way to add this option to the quiz settings form, then I think this would be a welcome option to add in Moodle core, so you are welcome to submit a feature request (and even a patch!)
(I say elegant, becuase the Quiz settings form is already complicated enough! I guess that, if we did this, it would also solve the issue where people don't want students to start the exam more than 15 minutes after the official start time.)