How to translate content in category.php or index.php with qtranslate?

Most correct way to do this would be to use WordPress translations.

You should replace this static text with:

<?php _e('YOUR TEXT', 'your_text_domain'); ?>

And add text domain to your theme.

More on this topic: http://codex.wordpress.org/I18n_for_WordPress_Developers

You can also…

… use qtrans_useCurrentLanguageIfNotFoundUseDefaultLanguage function.

Just use it like so:

<?php echo qtrans_useCurrentLanguageIfNotFoundUseDefaultLanguage('<!--en:-->Latest News<!--:--><!--fr:-->dernières Nouvelles<!--:-->'); ?>