[Vuejs]-Can I add a component in a method call inside a Vueitfy Calendar to display an icon on an event?

To format the event name you could use the event slot instead of event-name prop :

<v-calendar
          ref="calendar"
          :events="day_reports"
          :event-name="formatEventName"
        >
  <template #event="{eventParsed}" >
      <v-btn><v-icon>icon_name</v-icon> {{eventParsed.input.user.name}}<v-btn>
  </template>

</v-calendar>