by Ken Task.
With no designated directory and not using anything but defaults in moodle, backups go to moodledata/filedir/
Structure of moodledata/filedir/
Using a tool to browse files (moodle doesn't have such a tool) navigate to your moodledata/filedir/ First thing you need to do is read the 'warning.txt' file you will find there.
In filedir you will see a bunch of directories with 2 character/number names .. like 01 fg, ab, 02, 34, etc.
Inside each of those subdirectories, files whose name actually come from the DB (mdl_files table / contenthash column).
So a backup .mbz file might reside in:
moodledata/filedir/01/zz/ and it's name might be
01zzabunchofnumbersalphanumericcharacters.
The are of mimetype gzip (gunzip).
Without a plugin for backups the only way admins/teachers/manager see them is to visit a course and 'pretend' to restore.
If you install it, make sure you get the one compatible with your core moodle.
To see if you can find them another way .. use the mysql client on your server. Command line. Once connected via the client to your DB server and using the DB for Moodle (seen in config.php of your site), one can run this query:
select contenthash,component, filename,filesize from mdl_files where filename like '%.mbz';
If you don't have command line access, use whatever tool you have to work with the DB - in many hosting provider setups, that's phpmyadmin.
You've not mentioned how and where you are hosted. If on shared hosting, there could be caps/limitations that are the cause of the issue.
You might also review:
'SoS', Ken