Moodle Question:

Yes. Quizzes in Moodle are made from banks of questions that you create. You can then sort them into categories which can be 'published'.

By default all questions that you create will go into the "Default" category, but you can (and probably should) create new categories to organise your questions. A questionbank can be shared with other courses - this must be done by your Moodle administrator. If you wish to share questions with another moodle site, a category can be exported. There are a number of formats but the Moodle XML format is the most comprehensive. These can then be imported at the other site.