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

Go to cPanel --> Softaculous. Pick the preferred version of Joomla from Portals/CMS and click on the Install tab.

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 script/application that is compatible with your hosting account.

Log in to http://SiteGround.com/

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