You can use the
mysql-admin tools to connect and create scheduled backups of your databases to any location you desire, including your Ubuntu One folder. You can install
mysql-admin through Terminal using
sudo apt-get install mysql-admin or through a package manager like Synaptic.
mysql-admin tools have been deprecated and replaced by Oracle with MySQL Workbench which does not offer built-in backup options, but the older tools still function with current MySQL databases.
I'm performing essentially what you are trying to achieve using
mysql-admin, performing regular backups of my MySQL databases to GoogleDrive, and it works perfectly.