Filter the_content() in the Quote Post Format to Display Quotes

Try

$content = get_the_content();
$content="<span>"</span>".$content.'<span>"</span>';

echo apply_filters('the_content', $content);

CSS Solution:

<blockquote>
    <?php the_content(); ?>
</blockquote>

blockquote:before{
   content: '"';
}

blockquote:after{
   content: '"';
}