main.vue

<template>
  <div class="home-container">
    <Search v-if="filters" :filters="filters" />
    <Recommendations :dataRec="dataRec" />
    <TabCard :dataFav="dataFav" :topSellers="topSellers" />
    <ModalChat :profileChat="profileChat"/>
  </div>
</template>
<script>
import Vue from 'vue'
import Search from './Search.vue'
import Recommendations from './Recomendations.vue'
import TabCard from './TabCard.vue'
import Card from '../Card'
import ModalChat from '../ModalChat.vue'
export default {
  name: 'Main',
  components: {
    Search, Recommendations, TabCard, Card, ModalChat
  },
  data () {
    return {
      dataRec: null,
      dataFav: null,
      topSellers: null,
      profileChat: null,
      search: null,
      filters: null
    }
  },
  mounted () {
    this.profileChat = null
    this.getData()
  },
  methods: {
    async getData () {
      try {
        let jsonHome = (await Vue.http.get('/api/home/gethome')).body
        this.dataRec = jsonHome.recommendation
        this.dataFav = jsonHome.favourite
        this.topSellers = jsonHome.topseller
        this.filters = {
          bussiness: jsonHome.business,
          regions: jsonHome.region
        }
        console.log(this.filters)
      } catch (error) {
        console.error(error)
      }
    }
  }
}
</script>

search.vue

<template>
  <div class="mb-4">
    <h2 class="text-primary">Encuentre prestadores de servicios turisticos</h2>
    <div class="row">
      <div class="col-md-4">
        <multiselect
          v-model="bussines"
          :options="filters.bussiness"
          :multiple="false"
          track-by="id"
          label="name_es"
          placeholder="¿En que región desea buscar?"
          @input="clickData(bussines)"
        ></multiselect>
      </div>
      <div class="col-md-4">
        <multiselect
          v-model="region"
          :options="filters.regions"
          :multiple="true"
          track-by="id"
          label="name_es"
          placeholder="¿En que región desea buscar?"
          @input="clickData(region)"
        ></multiselect>
      </div>
      <!--
        <div class="col-md-4">
          <multiselect  v-model="states"
            :options="filters.regions"
            :multiple="false"
            track-by="id"
            label="name_es"
            placeholder="¿En que región desea buscar?">
          </multiselect>
      </div>
      -->
    </div>
  </div>
</template>
<script>
import Multiselect from 'vue-multiselect'
export default {
  name: 'Search',
  props: {
    filters: {
      type: Object,
      required: true
    }
  },
  components: {
    Multiselect
  },
  data () {
    return {
      bussines: null,
      region: null,
      states: null
    }
  },
  methods: {
    clickData (filterInput) {
      // const filterArray = Object.values(filterInput)
      // console.log(filterArray[0])
    }
  }
}
</script>

En tu main

<Search v-if="filters" :filters="filters" @update-filters="updateFilters" />

...
data() {
  return {
    selectedFilters: {
      business: null,
      region: null
     }
  }
},
methods: {
  updateFilters([filterName, filterValue]) {
    this.selectedFilters[filterName] = filterValue
  }
}

En tu search

<multiselect
          v-model="bussines"
          :options="filters.bussiness"
          :multiple="false"
          track-by="id"
          label="name_es"
          placeholder="¿En que región desea buscar?"
          @input="clickData('bussiness')"
        ></multiselect>
      </div>
      <div class="col-md-4">
        <multiselect
          v-model="region"
          :options="filters.regions"
          :multiple="true"
          track-by="id"
          label="name_es"
          placeholder="¿En que región desea buscar?"
          @input="clickData('region')"
        ></multiselect>
...
methods: {
  clickData(filterType) {
    this.$emit('update-filters', [filterType, this[filterType]])
  }
}