Moodle has to use moodledata/temp/backup/ to build the pieces to a backup ... files directory with associated files.xml, etc. It then has to zip them all up into an .mbz file. So the apache user needs to have full access rights to the moodledata/temp directory to be able to not only create files/folders, but copy the .mbz to another location in filedir by contenthash name, and then remove all files/folders it created to build the backup.
Sometimes, backups that fail leave files/folders and a .log file in moodledata/temp/backup/
That .log file larger than 0 bytes, might contain a clue.
From what you have described ... one quiz in one section of that course you suspect is the culprit. Then look for files that might be related in the /moodledata/temp/backup folder.
'spirit of sharing', Ken