So, you need to wrap the v-card in a template tag, and add on the v-card an v-if condition to show only the question index which equals to the currentIndex.

 <template v-for="(question, index) in quizQuestions">
    <v-card v-if="index === currentIndex" :key="index" class="elevation-0" height="62vh" style="overflow: scroll">
      <v-card-title class="text-h5 primary--text">Question {{ index + 1 }}</v-card-title>
      <v-card-subtitle>{{ question.value.questionTitle }}</v-card-subtitle>
      <v-card-text>
        <v-btn
          v-for="(answer, index) in JSON.parse(question.value.answers)"
          :key="index"
          class="elevation-0 mr-4 rounded-lg"
          @click="checkAnswer(answer.correct, index)"
          x-large
        >
          {{ answer.text }}
        </v-btn>
      </v-card-text>
      <v-row class="d-flex flex-column" no-gutters>
        <v-divider></v-divider>
        <div>Question {{ index + 1 }} of {{ quizQuestions.length }}</div>
        <v-progress-linear height="3vh" class="rounded-xl" :value="quizQuestions.length"></v-progress-linear>
      </v-row>
    </v-card>
  </template>

And in the js create a state for the currentIndex and a function to go to the next question by increase the currentIndex.

data: {
  return {
     currentIndex: 0,
  }
},
methods: {
  nextQuestion() {
    ++this.currentIndex
  }
}

You should add computed property to display the active question instead of your v-for to display all questions and it should look something like this

data: () => ({
  activeQuestionIndex: 1,
  quizQuestions: ["#f00", "#ccff00", "#eee", "#0f0"],
}),
computed:{
  activeQuestion: function () {
    return this.quizQuestions[this.activeQuestionIndex]
  }
}

So you remove v-for and everywhere where you used question.someproperty now you will use activeQuestion.someporperty

And activeQuestionIndex incrementation you should handle inside yours checkAnswer function