Air carrier Captain reported they were assigned a heading for weather avoidance and an altitude to maintain. After reaching the initial assigned altitude; ATC issued them a Low Altitude Alert and a climb.

Narrative

Operating to ZZZ; Runway XX. Tower amended the take-off clearance from FL230 to heading 280 at 10;000 ft. After take-off; the FO (First Officer) checked in and ATC adjusted the heading to 260 degrees for weather avoidance. We climbed to and leveled off at 10;000 ft. Captain was the PF (Pilot Flying) and it sounded busy so I turned the AP (Autopilot) on at 7;000 ft. to be sure that I caught all ATC transmissions. At 10;000 ft. ATC radio traffic got extremely busy as we were waiting for a clearance to a higher altitude. Then ATC gave us a Terrain Altitude alert; and a climbing turn with a heading. The FO tried to acknowledge and the Captain immediately turned and climbed. The aircraft indicated a single call-out of 2500 ft. once and as the aircraft was climbing/turning the lowest recorded Radar Altimeter was 2120 ft for 5 seconds. No aircraft damage; no terrain contact and no passengers were injured during the maneuvers. ATC reiterated the heading and climb but the ATC frequency was flooded with requests. We managed to acknowledge and then we were handed off to en-route ATC; and requested a further climb. Captain requested the ATC phone number after a few minutes.The Captain notified dispatch during the flight and pilot manager after landing. And then called and spoke with the Approach Control Supervisor and reviewed the situation. I gave the Supervisor the time to mark the tapes. He took my information and indicated that they will be calling me as they begin their investigation.