You can install Joomla on your account from the automatic script installation system - Fantastico. Here is the way of installing Joomla from Fantastico:

Go to cPanel --> Fantastico --> Joomla and click on New Installation.

For detailed instructions on how to install Joomla on your account, please check this tutorial.

You can also request the script to be installed on your account by SiteGround Technical Support Team. They would gladly assist you with the installation of any of the script/application that are compatible with your hosting account.

Log in to http://SiteGround.com/

Customer's Area --> Get Support Section --> HelpDesk --> Get script installed -> Script Installation