The first step is to create a set of template overrides for K2 (it's a file copy process). More here:
getk2.org/documentation/tutorials/templating-with-k2-and-the-concepts-of-sub-templates
Then edit the file category.php which should be in /templates/YOUR_TEMPLATE/html/com_k2/default/category.php and at the bottom of the file, add something like:
<?php if ($this->category->name == 'The name of that specific category'): ?>
<!-- Your custom HTML here -->
<?php endif; ?>
If you wish to check by category ID simply swap $this->category->name with $this->category->id.
