We've identified an issue with messaging during the upgrade of an existing site to Moodle 3.6.
Many thanks for reporting this Stefan Weber (https://moodle.org/mod/forum/discuss.php?d=379337)
Also discussed here: https://tracker.moodle.org/browse/MDL-64307
In summary; existing message conversations are no longer visible after the upgrade. Note that they are only hidden from view, the data is still present and in tact.
We plan to release a new minor version (3.6.1) to address this issue later today.
Please ensure to use this version for any upgrades.