Хотите получить предыдущее значение выбранной кнопки

$(document).on('click','.quantity-up',function(){
  $qtyElemnt = $(this).parents('.quantity').find('.form-qty');
  $qty = $qtyElemnt.val();
  $qtyElemnt.val(Number($qty)+1);
});

$(document).on('click','.quantity-down',function(){
  $qtyElemnt = $(this).parents('.quantity').find('.form-qty');
  $qty = $qtyElemnt.val();
  $qtyElemnt.val(Number($qty)-1);
});

.quantity {
   padding: 10px;
}
.quantity-nav{
    display: inline-block;
}
.quantity-button {
    display: inline-block;
    padding: 5px;
    background-color: #c7c5c5;
    border: 1px solid #585353;
    cursor: pointer;
}

<script src = "https://cdnjs.cloudflare.com/ajax/libs/jquery/3.3.1/jquery.min.js"></script>


<div>
  <div class = "quantity">
        <input type = "number" name = "qty" value = "1" class = "form-qty form-control" min = "1">
         <div class = "quantity-nav">
              <div class = "quantity-button quantity-up qty-up">+</div>
              <div class = "quantity-button quantity-down qty-down">-</div>
         </div>
    </div>
    <div class = "quantity">
        <input type = "number" name = "qty" value = "1" class = "form-qty form-control" min = "1">
         <div class = "quantity-nav">
              <div class = "quantity-button quantity-up qty-up">+</div>
              <div class = "quantity-button quantity-down qty-down">-</div>
         </div>
    </div>
    <div class = "quantity">
        <input type = "number" name = "qty" value = "1" class = "form-qty form-control" min = "1">
         <div class = "quantity-nav">
              <div class = "quantity-button quantity-up qty-up">+</div>
              <div class = "quantity-button quantity-down qty-down">-</div>
         </div>
    </div>
</div>

document.addEventListener("DOMContentLoaded", function() {
  const qtyWraps = document.getElementsByClassName('quantity');
  for (let i = 0; i < qtyWraps.length; i++) {
    const qtyWrap = qtyWraps.item(i);
    const input = qtyWrap.querySelector('.form-qty');
    const up = qtyWrap.querySelector('.qty-up');
    const down = qtyWrap.querySelector('.qty-down');
    const output = qtyWrap.querySelector('.output');
    up.addEventListener('click', function(e) {
      e.preventDefault();
      addValue(1);
    });
    down.addEventListener('click', function(e) {
      e.preventDefault();
      addValue(-1);
    });
    input.addEventListener('input', function() {
      output.textContent = input.value
    });
    const addValue = function(value) {
      const current = parseInt(input.value);
      const min = input.getAttribute('min') || -Infinity;
      const max = input.getAttribute('max') || Infinity;
      const step = input.getAttribute('step') || 1;
      const newValue = Math.min(max, Math.max(min, current + value * step));
      input.value = newValue;
      if (newValue <= min) down.setAttribute('disabled', 'disabled');
      else down.removeAttribute('disabled');
      if (newValue >= max) up.setAttribute('disabled', 'disabled');
      else up.removeAttribute('disabled');
      input.dispatchEvent(new Event('input'));
    }
    addValue(0)
  }
});

.quantity {
  display: block;
  width: 500px;
  margin: auto;
  text-align: center;
}

.quantity .form-qty {
  display: inline-block;
}

.quantity .quantity-nav {
  display: inline-block;
}

.quantity .output {
  background: yellow;
  width: 500px;
  margin: 1em auto 0;
}

<div class = "quantity">
  <input type = "number" name = "qty" id = "qty" value = "1" class = "form-qty form-control" min = "1">
  <div class = "quantity-nav">
    <button class = "quantity-button quantity-up qty-up">+</button>
    <button class = "quantity-button quantity-down qty-down">-</button>
  </div>
  <!-- I put it here to show the output result -->
  <div class = "output">1</div>
</div>