When copying a quiz or assignment to a new Canvas site, their setting in the new Canvas site is automatically set to the Published state. If your Canvas course is already Published, this means students may get a notification based on their notification preferences and will be able to access the quiz or assignment content immediately unless you quickly go in and set the item to Unpublished. 

Canvas sites/courses.

Automatic behavior by Canvas.

There are a few workarounds:

• By knowing this behavior, quickly go in and set the item to Unpublish as quickly as possible
• Copy all quiz/assignment content over before the Canvas site itself is set to Publish and made available to students, so even though the items are set as published, students still cannot enter the course to access them