Can I be notified when zones are opened or closed on my alarm system ?

Generally only if the zone openings and closings occur while the system is armed.

The system won't let you know if a zone is faulted or opened while the system is disarmed, however while armed, it would tell you if the zone was violated and returned to original state.

The reporting format your system is being monitored in has a lot to do with the amount of information that is readily available during an alarm transmission. Older systems and formats may simply provide an account number and an event, such as burglar alarm, while the newer formats, such as SIA and Contact ID can provide a wealth of information detailing the events as they occur.

In a local sense, when you are on the property, most systems offer a chime mode, which would notify via keypad when a protected door or window is opened. Some of the panels even offer a voice status or chime feature.

Also see Can I be notified when my system is armed or disarmed ?