Detecting delimiter on v-carousel being clicked

Something like this should work

<button @click="handleClick">...</button>

See event handling docs.

EDIT

assuming you're using the carousel component from vuetify, you could also use their events.

<template>
  <v-carousel v-model="model" @change="handleChange">
    <v-carousel-item
      v-for="(color, i) in colors"
      :key="color"
    >
      <v-sheet
        :color="color"
        height="100%"
        tile
      >
        <v-row
          class="fill-height"
          align="center"
          justify="center"
        >
          <div class="display-3">
            Slide {{ i + 1 }}
          </div>
        </v-row>
      </v-sheet>
    </v-carousel-item>
  </v-carousel>
</template>