C525 Captain reported after engaging the autopilot; the aircraft initiated a right-hand turn and deviated from the departure route. ATC was notified of a terrain alert and informed the flight crew of the deviation.

Narrative

While operating as the Captain for Company; an incident occurred during the execution of the ZZZZZ Departure Procedure with the ZZZ transition at ZZZ. Prior to departure; the First Officer and I reviewed the departure procedure and input the necessary data into the Flight Management System; FMS. However; upon reaching the end of the runway; the First Officer noticed a discrepancy in the FMS display. In the interest of safety; we requested a brief delay from Air Traffic Control; ATC; to review the departure procedure again. After confirming the accuracy of the information and receiving clearance from Tower; we commenced the departure. However; during the climb and after retracting the landing gear and flaps; upon engaging the autopilot; the aircraft initiated a right-hand turn that deviated from the prescribed departure route. ATC promptly informed us of the deviation and a terrain alert they had received. Assessing the situation and maintaining visual separation with terrain; we decided to continue the turn towards the ZZZ VOR while coordinating with ATC. We requested direct routing to the ZZZ VOR; but ATC assigned us a specific heading instead. The incident concluded with the completion of the assigned heading and ATC providing us with a possible deviation and contact information for further communication.Lessons learned from this incident include the importance of thorough preflight preparations; effective communication with ATC; continual situational awareness; and conducting post-incident evaluations to prevent similar occurrences in the future. Additionally; I recognize that in a situation where the autopilot initiates an unexpected deviation from the planned route; it is crucial to disengage the autopilot immediately. In this case; activating the VOR and flying the VOR track would have provided a reliable and known navigation reference while ensuring safe separation from terrain and other airspace. Once the FMS was corrected; the autopilot could have been reengaged to resume the intended route. This incident underscores the significance of remaining adaptable; prioritizing safety; and utilizing appropriate back-up procedures in the face of unexpected circumstances.