TRACON Controller reported a departing aircraft was not climbing quickly enough and entered an area below the minimum altitude. The Controller issued a low altitude alert and vectored the aircraft away from terrain.

Narrative

Aircraft X called for departure clearance on the ground at ZZZ. I issued the clearance to ZZZ1 via the ZZZZZ departure procedure. On his readback I didn't hear him say ZZZZZ departure so I verified it. After listening back to it was still a questionable readback as he added other things to it. I radar identified him after his check in. He then stated he was on a 160 heading then direct destination which concerned me because they were in a very slow climb approaching higher MVA's (Minimum Vectoring Altitude). I verified multiple times he was on the ZZZZZ departure and they said affirmative. I told them to give me best rate through 8000 [ft.] to get them above the next MVA ASAP. They then checked in level 8000 on their current heading (190); approaching a 8500 MVA. I Issued a climb to 13000 which was read back and then an expedited climb through 10000 as well as a low altitude alert. I then turned them away from terrain on a 020 heading and verified they were climbing after not observing them climb at all. I don't believe they were ever on the south valley 1 departure as they said they were; they entered a 8000 MVA at 6500 and a 8500 MVA at 8000. Readback; hear back was a contributing factor as the pilot had an accent. When clearances were readback by the pilot; they were not followed. Before releasing them; I should have been 100% sure they read back the departure procedure before releasing even after verifying it a second time with a clear readback. I should have paid more attention to every word they were saying and it would have clued me in sooner that they were unsure of the procedure.