VueJS cannot use MongoDB directly. VueJS is a frontend library (works in the browser), while MongoDB is a server-side database technology. In the course you mentioned, the author is probably bridging VueJS and MongoDB through an ExpressJS app, which runs in a server. You should be seeing things like fetch( or axios in the examples. Those words indicate the client-side application connecting to a backend.
The models are probably written for a MongoDB library that runs in ExpressJS. Controllers must be either self-developed or using a library that allows creating controllers for ExpressJS.
Notice I keep saying ExpressJS. There are other server frameworks for NodeJS, but ExpressJS is the most popular one.
Also, the only difference between running your own MongoDB instance and using MongoDB atlas (for the ExpressJS app) is where to connect. One connects to MongoDB running on http://localhost, the other connects to something like https://srv05-nickb.atlas.mongodb.com
Long story short, yes, you can use MongoDB with Vue or Quasar. But it won’t be Vue or Quasar using / connecting to MongoDB but an intermediary app written in something like ExpressJS.