Hi,
You'll have to edit the file "cart" of the view "checkout" via the menu Display>Views and replace the refresh button code by two HTML links with an onclick attribute which would have some javascript code to update the quantity field:
<a href="#" onclick="var qty_field = document.getElementById('hikashop_cart_quantity_<?php echo $row->cart_product_id;?>'); if (qty_field && qty_field.value != '<?php echo $row->cart_product_quantity; ?>'){<?php echo $input; ?> qty_field.form.submit(); } return false;" title="<?php echo JText::_('HIKA_REFRESH'); ?>">
<img src="<?php echo HIKASHOP_IMAGES . 'refresh.png';?>" border="0" alt="<?php echo JText::_('HIKA_REFRESH'); ?>" />
</a>