Hi,
The out of stock message will appear automatically on the product details page on the frontend once stock for the product goes to 0.
There is no option to not display it.
If you don't see it, either you've already have a view override (made on your end or by your template provider) which removes it, or you have a translation override of the text for it on the frontend which removes it, or you have custom CSS which removes it, or you have some page caching in place preventing you from seeing the change and you need to clear it.
The code to display it is in the view file product / quantity :
<!-- STOCK MESSAGE -->
<?php
$stock_class = ($stock != 0) ? "" : " hikashop_product_no_stock";
?>
<span class="hikashop_product_stock_count<?php echo $stock_class; ?>">
<?php
if(!empty($this->row->product_stock_message))
echo JText::sprintf($this->row->product_stock_message, $stock);
elseif($stock > 0)
echo (($stock <= $threshold_stock_message && JText::_('X_ITEM_IN_STOCK') != 'X_ITEM_IN_STOCK') ? JText::sprintf('X_ITEM_IN_STOCK', $stock) : JText::sprintf('X_ITEMS_IN_STOCK', $stock));
elseif(!$in_stock)
echo JText::_('NO_STOCK');
?>
</span>
<!-- EO STOCK MESSAGE -->
So either that code is missing or there is a problem with the translation key NO_STOCK