Plesk in Linux does not provide the option to generate backups for MySQL databases via Backup Manager. The only way to get a backup is to use the export option via phpMyAdmin. Please click here for details.
The only way to automate the backup process is to configure a CRON for the following script.
#!/bin/bash
# Set the datestamp, login credentials and backup directory
export date=$(date +\%Y\%m\%d)
export creds="-uadmin -p`cat /etc/psa/.psa.shadow`"
export backupdir="/mysqlbackup"
# delete week old files
find ${backupdir}/ -regex '.*.dump.gz' -mtime +4 -exec rm {} \;
# dump databases to the backupdir
echo "show databases;" | mysql ${creds} | egrep -v ^Database$ | \
awk '{print "mysqldump ${creds} "$1" | \
gzip > ${backupdir}/db-"$1"-${date}.dump.gz"}' | \
sh
As per the above script, backup will be created under the /mysqlbackup directory (create it manually on your server) and the name of the backup will be as db-dbName-yyyymmdd.dump.gz. Furthermore, upon execution the script will check for old backups and delete them automatically depending upon the value of mtime you want to specify.