Air carrier Captain turned toward mountainous terrain after takeoff and the flight crew received a terrain warning. The Captain continued the turn and departure with no correction made.

Narrative

Taking off out of ZZZZ; Runway XX. As the Pilot monitoring for this leg I was waiting for the passengers to arrive in the FBO. The Pilot Flying/Captain came in to use the restroom and I asked him if he looked over the departure and if he understood it. He replied yes and said it is pretty simple. Passengers arrive; load bags; we start engines; Etc. We take off and at 400 feet AGL I call 400 ft; he responds back with 'flaps up' and begins his turn. Instead of turning left he rolled into a right turn toward the mountainous terrain. I quickly say that we should be turning left and that I have never turned right taking off on this runway and that I don't think that the right turn is correct. He responds with 'Well that is what the GPS is telling me'. Shortly after we hear the Aural warning of the TAWS saying 'terrain terrain terrain'. Pilot flying continued the right turn to the next fix and we continue on the departure. My suggestion is make sure both pilot flying and pilot monitoring are both briefed thoroughly prior to takeoff. Another Action that could have been taken is to double check all of the fixes in the box prior to takeoff.