If Query In Sidebar

by

Assign the return value of the first check to a variable, and test if there is something in it:

if ( $quote = get_post_meta( get_the_ID(), 'FeaturedQuote', TRUE ) )
{
?><h3 class="featuredquote"><?php the_title(); ?></h3>
<blockquote><span class="bqstart">&#8220;</span><?php
    echo $quote;
    ?>
<span class="bqend">&#8221;</span></blockquote>
<?php
}
elseif ( $img = get_post_meta( get_the_ID(), 'InsertFullImg', TRUE ) )
{
    echo "<img src="https://wordpress.stackexchange.com/questions/98372/$img" alt="">";
}

Note, a <h3> cannot be the parent of a <blockquote>. Headlines can contain inline elements only.