There are several ways to contact maintenance while in flight.
One way is via a radio be it VHF, HF or satellite. Using this method you can either dial directly or use a phone patch*. Another way is via ACARS, which essentially is a text message system much like that used on mobile phones.
Either way, you will be in contact with your own maintenance.
*Phone patch: The ground station you are calling connects you to the person you want to talk to via a telephone connection.