Can I integrate LMS to Google App?

Yes, you can Integrate LMS to Google App, and one very good option for such LMS is Paradiso LMS. Leverage and share work files, documents and online training content across Google Apps with Paradiso Learning Management System. Some of its features are:
1. Google Apps LMS Single Sign-On
2. Enhanced Collaboration with Google Apps LMS Integration
3. Importing Files into LMS from Google Apps
4. User Auto-Enrollment

I’ve used Blackboard, Moodle and eCollege and none of them offer integration with Google Apps in the way you frame it.

We have a custom single sign-on that provides access to both Google Apps and the LMS but the Google Apps (for example Docs) do not work within the LMS (users are redirected outside the LMS).