B737-700 Captain reported a track heading deviation during takeoff resulting in a CFTT. The Captain stated the First Officer noticed the error and immediately corrected back on course away from the mountains.

Narrative

Departing ZZZ we taxied to Runway XX. We had completed all checklists as well triangles for the runway change they requested and was about to takeoff. ZZZ radio issued us a clearance with a void time of 4 mins past the hour. ATC then said; your cleared to depart 'direct ZZZZZ'. We placed ZZZZZ in L1; confirmed it; executed and departed. As soon as we lifted off the flying pilot (my FO) was still VMC and said this doesn't look right and took immediate correction away from the mountains back to the left. I clarified with Center that what they wanted was the ZZZZZ1 departure via the ZZZZZ transition and not direct ZZZZZ. Center agreed and we corrected back on course. We both had VSD up and had no terrain alerts or warnings on the VSD.Although it did not initially feel rushed looking back; we were at the end of the runway ready to go with many checklists and changes happening in a short period of time that seemed to be going quickly and smoothly. I think this gave us a sense of being on a roll and feeling caught up. If we had not been so close to departing; I may have stopped to look at my chart which would have caught my attention and prompted me to speak up before executing this issuance in the box.