###作成物の説明
atomics設計を参考にしています。
atoms→molecules→organisms→templates→pages
(孫)　　　（子）　　（親）　　今回のコンポーネントはこの対応で進めています。

###うまくいかないとき
(孫)


<template>
  <v-menu
    v-model="calendar"
    :close-on-content-click="false"
    :nudge-right="40"
    transition="scale-transition"
    offset-y
    min-width="200px"
  >
    <template v-slot:activator="{ on }">
      <v-text-field
        :value="date"
        v-on="on"
        :label="label"
        prepend-inner-icon="mdi-calendar-range"
        readonly
        crearable
      />
    </template>
    <v-date-picker
      v-model="date"
      @input="calendar = false"
      @change="$emit('input', $event)"
    />
  </v-menu>
</template>

<script>
export default {
  props: {
    label: {
      type: String,
      default: 'Not Defined'
    },
    value: {
      type: String,
      default: ''
    }
  },
  data: () => ({
    // date: new Date().toISOString().substr(0, 10),
    date: '',
    menu: false,
    modal: false,
    calendar: false
  }),
  watch: {
    value (val) {
      this.date = val
    }
  }
}
</script>

(子)


<template>
  <span class="calendar-style">
    <input-calendar
      :label="label1 +`from`"
      v-model="dateFromTo[0]"
      @change="$emit('input', dateFromTo)"
    />&emsp;
    <input-calendar
      :label="label1+`to`"
      v-model="dateFromTo[1]"
      @change="$emit('input', dateFromTo)"
    />
  </span>
</template>

<script>
import InputCalendar from '~/components/atomics/InputCalendar'

export default {
  components: {
    InputCalendar
  },

  props: {
    value: {
      type: Array,
      default: () => (['', ''])
    },
    label1: {
      type: String,
      default: ''
    },
    label2: {
      type: String,
      default: ''
    }
  },

  data () {
    return {
      dateFromTo: ['', '']
    }
  },

  watch: {
    value (val) {
      this.dateFromTo = val
    }
  }
}
</script>

<style lang="scss" scoped>
  .calendar-style {
    display: inline-flex ;
  }
</style>

(親)


<template>
  <span>
    <v-col cols="12" class="calendar">
      <v-row>
        <p>{{ arriveDay }}</p>
        <calendar-from-to
          :label1="arriveDay"
          v-model="FromTo"
        />
        <p>{{ shipDay }}</p>
        <calendar-from-to
          :label1="shipDay"
          v-model="S_FromTo"
        />
      </v-row>
    </v-col>
    <v-row>
      <v-col cols="3">
        <input-arrive-place-company
          v-model="companyName"
        />
      </v-col>
      <v-col cols="3">
        <input-arrive-place-company-kana
          v-model="companyNameKana"
        />
      </v-col>
      <v-col cols="3">
        <input-ware-house v-model="warehouse" />
      </v-col>
    </v-row>
    <v-row>
      <v-col cols="3">
        <input-warning
          v-model="alert"
        />
      </v-col>
      <v-col cols="3">
        <input-status
          v-model="status"
        />
      </v-col>
      <v-col cols="3">
        <Input-registered-person
          v-model="person"
        />
      </v-col>
    </v-row>
  </span>
</template>

<script>
import CalendarFromTo from '~/components/molecules/CalendarFromTo'
import InputArrivePlaceCompany from '~/components/atomics/InputArrivePlaceCompany'
import InputArrivePlaceCompanyKana from '~/components/atomics/InputArrivePlaceCompanyKana'
import InputWareHouse from '~/components/atomics/InputWareHouse'
import InputWarning from '~/components/atomics/InputWarning'
import InputStatus from '~/components/atomics/InputStatus'
import InputRegisteredPerson from '~/components/atomics/InputRegisteredPerson'

export default {
  components: {
    CalendarFromTo,
    InputArrivePlaceCompany,
    InputArrivePlaceCompanyKana,
    InputWareHouse,
    InputWarning,
    InputStatus,
    InputRegisteredPerson
  },
  data () {
    return {
      arriveDay: '納品日',
      shipDay: '出荷日',
      warehouse: [],
      companyName: [],
      companyNameKana: [],
      alert: [],
      status: [],
      person: [],
      date: '',
      FromTo: ['', ''],
      S_FromTo: []
    }
  },
  methods: {
  }
}
</script>

うまくいったとき

孫を直しました。
template内で@での呼び出しで、emitではなくonChangeというfunctionを呼び出し、$emitでinput とchangeを発火させるようにしました。
(孫のtemplate内)


<v-date-picker
      v-model="date"
      @input="calendar = false"
      @change="onChange"
    />

(孫のscript内）


methods: {
    onChange (e) {
      this.$emit('input', e)
      this.$emit('change', e)
    }
  }

孫のtemplate内の$emitの呼び出しだけではうまく子のvalueの値が書き変わらなかったので困っているところ、
助けてもらい、functionで強制発火する方法で解決できました。