Air carrier Captain reported ATC issued a low altitude alert during missed approach procedure at BUR.

Narrative

The arrival briefing for the ILS Z Runway XX was thorough and we were prepared for the likely missed approach. The visibility was reported above charted minimums. The ceiling was reported below the DH. I included in my briefing that I would fly the approach down to minimums with the autopilot on. Upon reaching our DH we did not see the runway and a missed approach was initiated. During the initial climb; I chose to disengage the autopilot. The procedure is to climb to 1300 ft. then a climbing right turn to 4600 ft. on a 210 heading. We believe we followed the correct procedure with accurate guidance from SRS and the programed go-around procedure. Upon switching from Tower to Departure; we were instructed to climb to 6000 which we did. At about 5700 ft. we were told that we were below the MVA of 6000 ft. for that area and in danger of hitting terrain and were told to climb to 7000 ft. Being in IMC in mountainous terrain; this triggered a reactive response in me. I feel like I responded similar to how I would a GPWS warning. This aggressive climb caused me to climb up to about 8000 ft. The matter was resolved quickly and we continued the flight to our diversion airport.Later; I called the Departure Controller to discuss what they claimed was a possible pilot deviation. I was told that we flew too far in the initial climb before making the right hand turn. This is what triggered the altitude alert on ATC's end. They also discussed with me the altitude deviation ft. when cleared to 7000 ft.Throughout the arrival and go-around; both pilots had Terrain on ND. We saw no GPWS or TCAS alerts.