TRACON Controller reported an aircraft deviated from departure procedure and entered an MVA. The reporter issued a low altitude alert and questioned the pilot's ability to maintain own terrain and obstruction clearance.

Narrative

I had released Aircraft X off of ZZZ around XA:17. Aircraft X checked in 2 miles north; northeast of ZZZ. I radar identified the aircraft and continued to work arrivals into ZZZ1. I issued traffic to Aircraft X regarding arriving aircraft. Aircraft X was expected to remain on a 120 heading and climb to 2100 ft. Arrival traffic on a south dog leg for ZZZ1 Runway XXR was restricted in descent to 3100 ft. At this time; I believe I issued traffic to Aircraft X in regards to the air carrier restricted above them. After continuing to control my arrival traffic and issuing speed restrictions; Aircraft X was observed turning due south. At this point I instructed Aircraft X to expedite their climb to 5000. I informed Aircraft X that they were supposed to have remained on the 120 heading as per the procedure. It appeared as if Aircraft X turned southeast at this point on their own. Aircraft X entered the 3000 MVA south of ZZZ and I issued a low altitude alert to them; dictating the MVA and to check their altitude immediately. Their altitude at this point was 1500 or 1600.At this point I got the Supervisor's attention and let them know that I had a ZZZ departure deviating from procedure departure and entering an MVA. I then began to question Aircraft X regarding their ability to maintain their own terrain and obstruction clearance. I had a hard time deciphering whether the pilot was stating that they 'can' or 'can't' maintain their own obstruction clearance. This went on for a moment and the aircraft was depicted descending to 1400 ft. as they entered the 3400 ft. MVA. Finally; the pilot seemed to state that they were VFR and watching the ground. I queried the pilot as to their intentions; 'Aircraft X do you wish to cancel IFR and proceed VFR?'. The pilot stated 'no.' I continued to insist that they expedite their climb to 5000 ft. The pilot came back a few times; with what sounded like a request for their filed altitude of 4000; and I continued to issue a climb to 5000 ft. When the aircraft entered the 4000 ft. MVA and they were observed leaving 4000 ft. I issued them a vector to 160 degrees for vectors to ZZZZZ; as they were inbound to ZZZ2. This seems to be a common event recently and in the past; the aircraft must be made to understand the instructions on the departure procedure. I also believe a catalyst to this ongoing issue is the confusion caused by an IFR departure procedure that requires aircraft to depart and fly 'VFR' and remain VFR under 1100 ft. until crossing the ZZZ3 radial.