We do not have a formal mentorship program, but The Document Foundation consist of friendly people, who can help you make your first contribution. We have experts in all parts of the code/documentation/QA etc see Find the expert
Once you decide what area of the project you are interested in contributing to and explore the information available on this wiki, you can introduce yourself to our main mailing list libreoffice, normally you will get a respond fast with an offer to help you.
Some of the people in The Document Foundation devote time to help new people getting started, in a coach style. This effort is being coordinated by Shinnok. Feel free to mail comments / questions directly.
The coach/mentor can help you build the code, identify an easy bug to start with in the Easy Hacks list, and help you with your patch for that bug. The mentor can guide you through your subsequent contributions and point to the resources for solving a particular task.
We use connect via webchat for online fast communication, this is especially usefull if you are stuck with a problem and would like a short advice.
Please see Find the expert for a list of people to contact with specific problems.