Both folders and quizzes can be moved to different folders. To move quizzes/folders, first go into the Quizzes tab.
Locate the folders or quizzes to be moved and select the item(s) by clicking the checkbox to the left of the name and then click Move.
Choose the folder you'd like to move the quizzes/folders to. When choosing a folder, the menu will navigate into the folder:
You can choose Move Here if you're satisfied with the location. You can also choose Create a Folder to create a subfolder.
Enter the name of the subfolder and click the check mark icon. Once created, you can click Move Here to move your quizzes/folders over.