This problem is most commonly caused by bad sample rate conversion within the editing application, FCP in your case. Check your source audio sample rate and compare that to the settings for the FCP sequence/project/clips.
I’ve never had anything go out of sync exported out of QT Pro in over a decade, but I have with cleaner, years ago. Cleaner hasn’t been update usefully in years and I wouldn’t recommend it to anyone anymore. QT Pro is a very niceplayer with export features but I wouldn’t use it for serious transcoding/compression work. Have a look at Procoder from Grass Valley. is you want a high quality, professional and regularly updated application: https://www.canopus.com/products/ProCoderSW/index.php
It should be easy to get both to look good, but without having any details of what you are working with (Video format, NLE etc) and what you are delivering to (format, audience, bandwidth etc)it’s impossible to offer any advice or tips.