You can transfer quiz questions between modules, so long as you have editing teacher rights in both.

This refers to Moodle 3.6 from July 2019 onward

How to: transfer questions for quizzes between Moodle modules

Exporting the questions from a module

  1. Go into the module with the questions you want to export.
  2. Drop down the Cogwheel Edit   menu 
  3. Click on More...
  4. Scroll down to  Question bank > Export

  5. Select the Moodle XML format

  6. Click on the Export questions to file button  
  7. Your export file should start to download. If it doesn’t, click as indicated on the download page to download the file.
  8. Save the file to a folder on your hard drive (it may go into Downloads by default)

Importing the questions into another module

  1. Go into the module in Moodle into which you want the questions added.
  2. Drop down the Cogwheel Edit   menu 
  3. Click on More...
  4. Scroll down to  Question bank > Import
  5. Select the Moodle XML format.
  6. EITHER:
  7. Click the Import button
  8. Then click Continue to complete the process of importing the questions