Air carrier First Officer reported an ATC low altitude alert and a terrain caution while on the initial approach in mountainous terrain. The pilot leveled off and started a climb when the terrain warning stopped and the flight continued to a safe landing.

Narrative

Date; Flight XXX from ZZZ to SJC. Momentary Terrain Caution on visual approach 30L SJC. We descended via the BRIXX4 Arrival and received additional clearance to 7;000 ft. We had loaded; briefed; and were expecting the RNAV (RNP) Z 30L approach at JILNA due to mountainous terrain and it would be a night approach. Approach Control denied the RNAV Approach and told us we would be too high; so they gave us a vector between KLIDE and HIVAK with a descent to 5;100 ft. and said expect the visual approach 30L. Captain continued the descent and asked me; as the PM; to load the ILS 30L. Captain picked up the airport visually and Approach cleared us for the visual approach. Since we were high on the approach; Captain then started a descent and set 2;700 ft. for HIVAK in the ALT window with Level Change and began to configure. At or about (time); we both noticed our descent rate was a little high over the mountains and Captain began to level off and climb slightly starting at around 4;300 ft. MSL. As he was climbing; Approach advised that Min Safe in that area was 4;700 ft. and they got a low altitude warning. We acknowledged. Almost simultaneously we got a momentary Terrain Caution on the display. We both noticed and continued our climb and immediately the Terrain Caution went away. The rest of the approach and landing were uneventful. Solution: Lesson learned. RNAV Z 3OL was the requested and the desired approach for the reasons mentioned above. We were also aware that we were high on the approach due to the vector. Instead of the loading the ILS 30L and using Level Change for the visual approach we should have loaded the RNV Y 30L; recruised our ALT and used VNAV and LNAV for our approach. Additionally; we could have asked for ATC for radar vectors for the approach.