Canvas does not have a native test monitoring function, however there is a whole industry built around locked down browsers and proctoring tools that integrate with learning management systems. If you search here in the community you will find, among others, information about ProctorU and Respondus Lockdown Browser.
I hope this helps you!
Would it work if I require students to connect to a meeting on Zoom (cameras on) and then have them start the quiz on Canvas?
Yes, you could have each student log into Zoom, share their screen and then take the quiz while you watched although that would obviously be labor intensive. If you are dealing with students unexpectedly out of school because of school shutdowns could you perhaps entrust someone else like a parent to act as proctor?
@tajordan , one of our awesome Canvas Coaches just wrote a blog post that compiles his podcasts on using Zoom for various purposes, including proctoring: https://community.canvaslms.com/groups/admins/blog/2020/03/25/academic-coronavirus-preparedness-podc...