Hi there, @sara_samples1 ...
Yes, this is possible. In fact, I just tested it out in my own sandbox course, and it worked as expected. On your "Quizzes" page of your course, find the Classic Quiz that you want to make a copy of. Then, on the far right-hand side of the Classic Quiz name, click on the three-dot kebab icon. Choose "Copy to...". You'll be prompted to search for a course. What you can do here is look in the address bar at the top of your browser, and you'll see something like:
https://SchoolName.instructure.com/courses/12345/quizzes
Your own CourseID number will be listed in that URL. Type in that number in the "Select a Course" box at the top of the fly-out menu that appears when you select the "Copy to..." option that I referenced earlier. This should then allow you to select the same course that you're already logged into. Click on the "Copy" button.
You can then go to your "Settings" page >> "Import Course Content" link, and under the "Content imports" heading, you should see a new row for this quiz copy. Once it is done, go back to your "Quizzes" page, and you should see it listed there.
I hope this helps a bit. Sing out if you have any questions about this. Thanks!