@manacrox your code saved my day! But there are a few lines I added to make it perfect for me and would like to share it back in case somebody finds it useful.
The “interesting” part are the masks set both on the input and the calendar. This makes it show and save only the month and year part.

      <q-input
        readonly
        input-class="cursor-pointer"
        label="Inicio"
        :value="e.periodo.inicio"
        @click="$refs.monthPicker.show()"
        mask="##/####"
        fill-mask="##/####"
      >
        <template v-slot:append>
          <q-icon name="event" class="cursor-pointer">
            <q-popup-proxy ref="monthPicker" transition-show="scale" transition-hide="scale">
              <q-date
                minimal
                mask="MM/YYYY"
                emit-immediately
                default-view="Years"
                v-model="e.periodo.inicio"
                @input="checkValue"
              />
            </q-popup-proxy>
          </q-icon>
        </template>
      </q-input>