How to unit test Vue component that append DOM-element to HTML body?

In your component definition, add a ref attribute to the content of the menu, in your case, the <v-card> component:

<v-menu>
  <v-btn slot="activator"></v-btn>
  <v-card ref="menu"></v-card>
</v-menu>

Then you'll be able to access that <v-card> component via wrapper.vm.$refs.menu in your test cases.