Air taxi pilot reported receiving a low altitude alert from ATC during departure climb due to flight toward rising terrain. Flight crew turned away from terrain and continued flight.

Narrative

Aircraft X was assigned a flight from ZZZ to ZZZ1; current conditions warranted takeoff to the south on runway XXL/R. Conditions were VFR at departure; enroute; and arrival destinations. Upon arrival at the aircraft; the PIC and pilot flying; began setting up the avionics inside the aircraft while myself (SIC and pilot monitoring) began my preflight outside of the aircraft. Once PIC completed setting up the avionics inside; we switched roles and I began to review the information that was put into the plane (Flight plan; altitudes; etc.) PIC filed the flight plans so he was the one receiving the pre-departure clearance. At the time; I had yet to receive a pre-departure clearance (PDC) so the flight plan I looked at was the ForeFlight cleared and expected routing. Later; we found that PIC attempted to send me a PDC but my email had logged out for security reasons and never got a notification nor did PIC tell me he sent it. We loaded into the aircraft and received taxi instructions to depart runway XXL; that ended up changing at the last minute to XXR. During checklist items PIC checked flight instruments which included saying what altitude we were cleared to and departure runway. We departed runway XXR and at a safe altitude PIC called for Autopilot to be engaged by myself. After AP was engaged; PIC selected NAV mode from heading mode and went direct to the first fix on our filed flight plan; which was approximately a 180 heading. We checked in with departure stating altitude leaving and altitude climbing. Around 9;500 feet MSL; departure alerted us to low altitude and gave us a turn to the left due to climbing terrain and advised us the lowest altitude in the area is 10;000 ft. We complied with instructions and continued to 12;000. The controller then told us go direct to our first fix ZZZZZ which was very close and at our 4 o'clock. In the turn the controller asked if we were turning directly to ZZZZZ and we responded we were but it was very close so we were trying to turn quickly. We then asked for another fix that was more in front of us and on the approach; the request was granted. The rest of the flight was uneventful and landing was completed safely. There was a breakdown in communication prior to departure and the PDC was not reviewed. This caused us to turn on course prematurely and without clearance. We were flying approximately a 180 heading while the departure needed us flying a 168 heading.During this event; we had no loss of separation from other aircraft nor were we in a position of compromised safety due to the terrain. The rapidly rising terrain was to our west and we had visual contact with the mountains. ForeFlight shows that we were approximately 4.5 miles East of the tallest peak. No Enhanced Ground Proximity Warning System (EGPWS) activation was noted.