Edit in JSFiddle

  const vm = new Vue({
    el: "#app",
    data: {a: ""},
    methods: {
      g: function() {
        this.a = Math.floor(Math.random() * 100000000000000000).toString(36)
      },
    },
  })
<div id="app">
  <div class="form-horizontal">
    <div class="form-group">
      <label class="col-md-2 control-label">パスワード</label>
      <div class="col-md-10">
        <input class="form-control" v-model="a" />
        <div class="help-block">
          <div class="btn btn-default btn-xs" @click="g">
            パスワード生成
          </div>
        </div>
      </div>
    </div>
  </div>
</div>