Getting bootstrap vue pagination to play with REST api

You need to set the per-page prop to 0 on the b-table component to disable the local pagination and allow b-pagination to handle the data. Here's an example:

new Vue({
  el: '#app',
  data() {
    return {
      items: [],
      fields: [{
          key: 'postId',
          label: 'Post ID'
        },
        {
          key: 'id',
          label: 'ID'
        },
        {
          key: 'name',
          label: 'Name'
        },
        {
          key: 'email',
          label: 'Email'
        },
        {
          key: 'body',
          label: 'Body'
        }
      ],
      currentPage: 0,
      perPage: 10,
      totalItems: 0
    }
  },
  mounted() {
    this.fetchData().catch(error => {
      console.error(error)
    })
  },
  methods: {
    async fetchData() {
      this.items = await fetch(`https://jsonplaceholder.typicode.com/comments?_page=${this.currentPage}&_limit=${this.perPage}`)
        .then(res => {
          this.totalItems = parseInt(res.headers.get('x-total-count'), 10)

          return res.json()
        })
        .then(items => items)
    }
  },
  watch: {
    currentPage: {
      handler: function(value) {
        this.fetchData().catch(error => {
          console.error(error)
        })
      }
    }
  }
})
<link type="text/css" rel="stylesheet" href="//unpkg.com/bootstrap/dist/css/bootstrap.min.css" />
<link type="text/css" rel="stylesheet" href="//unpkg.com/bootstrap-vue@latest/dist/bootstrap-vue.css" />
<script src="https://cdnjs.cloudflare.com/ajax/libs/vue/2.5.22/vue.js"></script>
<script src="//unpkg.com/babel-polyfill@latest/dist/polyfill.min.js"></script>
<script src="//unpkg.com/bootstrap-vue@latest/dist/bootstrap-vue.js"></script>

<div id="app">
  <b-table show-empty :items="items" :fields="fields" :current-page="currentPage" :per-page="0"></b-table>
  <b-pagination size="md" :total-rows="totalItems" v-model="currentPage" :per-page="perPage"></b-pagination>
</div>

You can also disabled local pagination in the table, so that your items provider becomes responsible for controlling the pagination.