They usually get saved back to where the original file is located. And usually Lightroom sees this file. If you save into original file loaction, LR does not see the file? You could also syncronize the LR folder.
I use a synology NAS which is virtually identical to a QNAP one and just tried this and it works normally. Saved file shows up in Classic correctly. No save as dialog. Simply saves under the -Edit standard name. I would check if the permissions are set right and if you see that you have read and write permissions on the folder the images are in. This is Classic 10.2 on Big Sur. I always use smb to connect to the NAS as it is faster than AFP but it actually doesn't matter.
P.S. I suddenly remember an issue with saving from photoshop to a NAS when the folder names on the NAS had special characters or when the file name has special characters in it. Do your file or folder names have any characters that are non-ASCII? Especially deadly are colons, commas, dollar signs, ampersands and hashes, and diacriticals are problematic too.