I finally found a solution that works. In case you need it, here it is:
<?php
if (has_post_format('quote', $post->ID)) {
$content = trim(get_the_content());
// Take the first quote from the content
$quote_string = extract_from_string('<blockquote>', '</blockquote>', $content);
// Make sure there's a quote on the content
if (!$quote_string == "") {
// Get the first quote and show it on the Loop
echo $quote_string;
} else {
// If not, show nothing
}
}
?>