There are several instructions on how to use the KeePass2 trigger mechanism to create a backup of your password database when saving the database.  However, all of the instructions I found were for Windows.  It took a bit of figuring out what is the proper configuration on Linux, so here are the necessary steps:

1. Select Tools → Triggers…
2. Click Add…
3. Type the name Backup database on save and make sure Enabled and Initially on are checked
4. Under the Events tab click Add…, select Saving database file and click OK (ignore the conditions)
5. Under the Actions tab click Add…, select Execure command line / URL and type in the following:
   File/URL:  cp
   Arguments:  "{DB_PATH}" "{DB_PATH}.{DT_SIMPLE}"
   Wait for exit:  checked
6. Click OK a few times

This will create a backup of the database file every time you save it (before the save).  The backup will be in the same directory as the original, with the current timestamp appended to the name.  If you prefer to backup to a different directory, use "/path/to/backups/{DB_BASENAME}.{DT_SIMPLE}" as the second argument instead.