Light Transport; Low Wing; 2 Turbojet Eng
Flight Crew Air Transport Pilot (ATP); Flight Crew Instrument; Flight Crew Multiengine
Flight Crew Last 90 Days 60; Flight Crew Total 11000; Flight Crew Type 600
Situational Awareness; Time Pressure; Workload; Communication Breakdown
ATC Issue All Types; Deviation - Altitude Excursion From Assigned Altitude; Deviation / Discrepancy - Procedural Published Material / Policy; Deviation / Discrepancy - Procedural Clearance; Inflight Event / Encounter CFTT / CFIT
Air Traffic Control Issued Advisory / Alert; Flight Crew Took Evasive Action
Human Factors; Environment - Non Weather Related
Part 91 flight [to] ZZZ with no passengers. Was descending on the [STAR] arrival into ZZZ. Reached the final fix of the arrival (ZZZZZ) for radar vectors at 8;000 ft. TRACON Controller gave us a heading of 190; descend to 6;300 ft. Made the turn to 190 and started down at 1;000 fpm; then at 7;300 ft. the Controller gave command of 'alt alert; turn immediately to a heading of 340 and climb to 8;000 ft!'  We made the turn and climb. On the original vector; we noticed on the assigned heading we where aimed towards [canyon] which has a peak of 5;720 ft.
Air taxi Captain reported while on approach and receiving a heading and altitude instruction from ATC; shortly thereafter receiving a terrain alert from ATC.