How to handle Paste(Ctrl+v or with mouse) event in vue.js?

You can simply use the paste event:

<textarea @paste="onPaste"></textarea>

...
  methods: {
    onPaste (evt) {
      console.log('on paste', evt)
    }
  }
...

It’s not a vue-specific event. See https://developer.mozilla.org/en-US/docs/Web/Events/paste