Nuxt

HTML

<link rel="stylesheet" href="https://maxcdn.bootstrapcdn.com/bootstrap/4.0.0/css/bootstrap.min.css" integrity="sha384-Gn5384xqQ1aoWXA+058RXPxPg6fy4IWvTNh0E263XmFcJlSAwiGgFAW/dAiS6JXm" crossorigin="anonymous">

<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/3.3.1/jquery.min.js"></script>
<script src="https://cdnjs.cloudflare.com/ajax/libs/popper.js/1.12.9/umd/popper.min.js"></script>
<script src="https://maxcdn.bootstrapcdn.com/bootstrap/4.0.0/js/bootstrap.min.js"></script>
<div id="app">


<div class="dropdown">
    <button class="btn btn-secondary dropdown-toggle" :style="test ? clickedBtn : null" type="button" id="dropdownMenuButton" data-toggle="dropdown" aria-haspopup="true" aria-expanded="false" @click="testing()">
        Dropdown button
    </button>
    <div class="dropdown-menu" aria-labelledby="dropdownMenuButton">
        <a class="dropdown-item" href="#">Action</a>
        <a class="dropdown-item" href="#">Another action</a>
        <a class="dropdown-item" href="#">Something else here</a>
    </div>
</div>
</div>

Vue

new Vue({
  el: "#app",
  data: {
    test: false,
    clickedBtn: {
      background: "blue",
      color: "white"  
    }
  },
  methods: {
  	testing(){
    	this.test = !this.test
      console.log('hasan')
    }
  }
})