[Vuejs]-Vue3: How do I access a function inside the methods attribute from a component's setup() method?

I’d recommend you to create a composable function. Define all your functions and state of the app like this:

 // /src/composables/useQuestions.js 
 import {ref, onMounted, computed} from 'vue'

 const questionArray = ref([])
 const currentQuestion = ref(0)
 const selectedAns = ref(null)
 const computed_answer_options = computed(() => {
    const questionObj = questionArray.value[currentQuestion.value]
    if (questionObj) return questionObj.options
    return []
 })
 const computed_question = computed(() => {
    const questionObj = questionArray.value[currentQuestion.value]
    if (questionObj) return questionObj.question
    return null
 })

 export default () => {
    onMounted(() => {
       window.addEventListener("keyup", (event) => {
          if (event.code == 'ArrowDown') {
             nextQuestion();
          }
          else if (event.code == 'ArrowUp') {
             previousQuestion();
          }
       });
    })

    // here you can call function that detect key releases and calls storeAnswer
    // I did't finish logic, but i think you got the point

    return {
       questionArray,
       currentQuestion,
       nextQuestion,
       previousQuestion,
       computed_answer_options,
       computed_question,
       storeAnswer
    }
 }

 const nextQuestion = () => {
    if (currentQuestion.value < questionArray.value.length - 1) {
       currentQuestion.value++;
    }
 }
 const previousQuestion = () => {
    if (currentQuestion.value > 0) {
       currentQuestion.value--;
    }
 }
 const storeAnswer = (ans) => {
    selectedAns.value = ans 
 }

in your components use like this:

 // App.vue, QuestionCard.vue
 ...,
 setup () {
   const {
       questionArray,
       currentQuestion,
       nextQuestion,
       previousQuestion,
       computed_answer_options,
       computed_question,
       storeAnswer
    } = useQuestions()
   questionArray.value // returns []
 }

I managed to solve my problem thanks to @Estus Flask, who pointed out my misusage of the Vue’s Composition API with Options API in my code.

Here’s the working code.

<script setup>
import { ref, onMounted } from 'vue';
import QuestionCard from "./components/QuestionCard.vue";


const questionArray = ref([
   {
      id: "123",
      question: "Which of these is a colour in the rainbow?",
      options: [
         'brown', 'red', 'black',
      ],
      selectedAns: null,
   }, {
      id: "456",
      question: "How many continents does Earth have?",
      options: [
         1, 7, 6, 9
      ],
      selectedAns: 6,
   }, {
      id: "789",
      question: "Which of these is a prime number?",
      options: [
         7, 4, 44, 
      ],
      selectedAns: 4,
   },
]);

const answerOptions = ref([]);
const currentQuestion = ref(0);


// methods to use in the App component -----------------------------------------------------------------
function nextQuestion() {
   if (currentQuestion.value < questionArray.value.length - 1) {
      currentQuestion.value++;
      setAnswerOptions();
   }   
}
function previousQuestion() {
   if (currentQuestion.value > 0) {
      currentQuestion.value--;
      setAnswerOptions();
   }
}
function setAnswerOptions() {
   answerOptions.value.length = 0;

   for (let i=0; i < questionArray.value[currentQuestion.value].options.length; i++) {
      answerOptions.value.push(
         {
            key: String.fromCharCode(65+i),
            value: questionArray.value[currentQuestion.value].options[i]                  
         }
      );
   }
}


// lifecycle hooks--------------------------------------------------------------------------------
onMounted(() => {
   // console.log('App mounted, setting up listeners!!');

   setAnswerOptions();

   window.addEventListener("keyup", (event) => {
      if (event.code == 'ArrowDown') {
         nextQuestion();
      }
      else if (event.code == 'ArrowUp') {
         previousQuestion();
      }
      
      const i = answerOptions.value.findIndex(element => element.key == event.key.toUpperCase());
      if (i > -1) {
         // console.log(`You have seleced ${answerOptions.value[i].value}, moving to next question`);
         questionArray.value[currentQuestion.value].selectedAns = answerOptions.value[i].value;
         nextQuestion();
      }
   });   

   // console.log('listeners all set!')
});
</script>


<template>
   <div>
      <div id='top_bar'>
         <button @click=" previousQuestion ">Previous Question (Up key)</button >
         <button @click=" nextQuestion ">Next Question (Down key)</button >
      </div>

      <div id='question_section'>
         <QuestionCard 
            :id=" questionArray[currentQuestion].id "
            :question=" questionArray[currentQuestion].question "
            :answer_options=" questionArray[currentQuestion].options "
         ></QuestionCard>
         
         <p v-if=" questionArray[currentQuestion].selectedAns != null ">
            {{ 'Your last answer was: ' + questionArray[currentQuestion].selectedAns }}
         </p>
         
         <div id='answer_options_section'>
            <ul>
               <li v-for=" option in answerOptions ">
                  <button :class=" {'selected-option': option.value == questionArray[currentQuestion].selectedAns } " >
                     {{  option.key + ') ' + option.value }}
                  </button>
               </li>
            </ul>
         </div>
      </div>
   </div>
</template>

QuestionCard.vue

<script setup>
import { ref } from 'vue'

// define the props used in this component
const props = defineProps({
   id: String,
   name: String,
   question: String,
   answer_options: Array,
});
</script>


<template>
   <div>
      <h5>Question ID: {{ id }}</h5>
      <h4>{{ question }}</h4>
   </div>
</template>