[Vuejs]-Vue app custom scss file – change font across entire app?

What you trying to do can be achieved by importing your google font inside of your <style> tag, in best case, in App.vue, your main component.

@import url('https://fonts.googleapis.com/css?family=Montserrat&display=swap');

Then you can use it like this. Please not to not scope this attributes:

html, body {
  font-family: 'Montserrat', sans-serif;
}

#app {
  font-family: 'Montserrat', sans-serif;
}