Moodle allows faculty to modify test taking attributes for individual students.
Open/Close Time Modification for an Individual
Go to your exam, and click on the Settings gear -> "User overrides"
Click "Add user override"
Search for the student's name who needs a modification to the open/close time of the exam.
You can modify the "Open the quiz" and/or "Close the quiz" time for the student, depending what you'd like to accomplish.
Click "Save" near the bottom of the window.