While writing the question in SE I already found the solution:
add_filter( 'get_the_excerpt', 'get_custom_the_excerpt' );
function get_custom_the_excerpt($excerpt) {
$post = get_post();
$format = get_post_format( $post );
// modify excerpt if it is a quote, for example set title as author
if ($format == 'quote') {
$output="<q>". $excerpt. '</q>
<p class="quoted">'.get_the_title().'</p>';
}
// return excerpt without change
else {
$output = $excerpt;
}
return $output;
}
For details check get_the_excerpt() and get_post_format().