G450 Captain reported ATC issued them a routing below the Minimum Vectoring Altitude resulting in a Ground Proximity warning.

Narrative

Departure from ZZZ; field elevation 5673'. Tower closed; non-controlled take off runway XXR; no moon. Field MSA 14;800'. First fix on filed flight plan: ZZZZZ; 235 degrees for 85nm. Clearance from ZZZ Approach was 'cleared as filed ZZZ1; climb to FL230…'; no SID assigned. We therefore briefed and planned textual ODP: initial takeoff heading 350 to 113 deg. Then; regardless of initial on course heading; turn right 142 degrees; intercept 280 radial to ZZZ2 VOR and climb in holding till above 16;500'; then on course. Waypoint for ZZZ2 VOR was loaded to facilitate the textual ODP (Obstacle Departure Procedure).Release instructions from ZZZ approach were to climb heading 350. Takeoff performed; flaps raised at 1500' AGL (roughly 7100' MSL) when ZZZ approach directed: 'left turn direct ZZZZZ.' Left turn commenced with trepidation; which immediately resulted in a solid RED display on GPWS; solid red GPWS display across entire PFD (Primary Flight Display) …inside of 25 nm ring. We were still just climbing thru 10;000' MSL. So; without asking permission; we exercised urgent authority; clicked off auto-pilot and immediately commenced a climbing right turn to avoid impending terrain impact. ZZZ2 was then notified of our deviation from their vectors. Flight to destination completed without any further incidents. ZZZ approach ATC tried to vector us into mountains below MVA; 6000' below MSA.Suggestions ZZZ approach ATC issue to correct.