I came up with this in my 3.7 sandbox site (to place in your theme's additional css):
/* Bring Quiz Timer to Top of Block */
#page-mod-quiz-attempt #mod_quiz_navblock .card-text.content.mt-3 {display: flex; flex-direction: column;}
#page-mod-quiz-attempt .qn_buttons {order: 2;}
#page-mod-quiz-attempt .othernav {order: 1; display: flex; flex-direction: column;}
#page-mod-quiz-attempt #quiz-timer {order: -1;}
/* Bring Quiz Timer to Top of Block */
#page-mod-quiz-attempt #mod_quiz_navblock .card-text.content.mt-3 {display: flex; flex-direction: column;}
#page-mod-quiz-attempt .qn_buttons {order: 2;}
#page-mod-quiz-attempt .othernav {order: 1; display: flex; flex-direction: column;}
#page-mod-quiz-attempt #quiz-timer {order: -1;}
I went to the Moodle Demo site to test this out and noticed that, there, they have a floating timer that stays at the top of the screen as you scroll down!
Check it out here: https://sandbox.moodledemo.net/mod/quiz/view.php?id=5
It's Boost Theme in Moodle 3.10
