I think you might want to give the Safe Exam Browser a try - http://docs.moodle.org/26/en/Safe_exam_browser .
At the very least you might want to turn on Browser security in the Extra restrictions on attempts section of the quiz administration settings. Double check though if a quiz navigates correctly with the browser security on - on Moodle 1.9 we had issues with the scroll wheel on the mouse etc in quizzes with multiple questions on one page - only up and down arrows worked after first clicking on some white space on the quiz page.