I am using Webmail (RoundCube) on cPanelX, using BlueHost web hosting.
I am planning to move to Google Apps, is there any way to move my existing mail to Gmail?
Since RoundCube supports IMAP, I searched for some solutions to migrate emails to Google using IMAP, and I did find imapsync. There are some steps available to guide you through (though it is mainly tailored for Linux users.
If it is not feasible for you, you can read more about migrating to Google Apps to see if any of these fits you.
BlueHost already have a Knowledgebase Article about How to access your e-mail with Gmail. You can skip the Outgoing Mail (SMTP) section as you presumably won't be using their service to send mail any more.
These are basically the instructions for using Google's Mail Fetcher to pull in old email, but they include BlueHost-specific settings. BlueHost had decent screenshots with their instructions, but for completeness they're as follow: