Air carrier First Officer reported receiving a low altitude alert from ATC on a visual approach to SAT airport.

Narrative

During vectors for a visual approach to 13R at SAT; we were vectored off the QUERVO1 arrival in the vicinity of TRTILA and assigned a descent to 3000 feet. Approaching 3000 feet we were then given a heading to intercept the final approach course and an altitude that I heard and read back as 2000'. The CA flying set 2000' and I confirmed the altitude. As we were turning on to final at approximately 2500 feet the Controller gave us a low altitude alert and instructions to climb to the MVA of 2800 feet and stated that he assigned us 3000 feet. The lowest altitude I saw as we started to climb was 2300 feet. We climbed to 2800 feet which put us above the glide path and we were cleared for the visual approach. The CA adjusted the approach path with a rate of decent that was higher than normal to get on glide path in time to put us in position for a stabilized approach. The landing as uneventful. The threats and errors were either an improper altitude assignment by ATC or a misheard altitude; by myself and confirmed by the CA; and an incorrect read back not caught by ATC. Also; my loss of situational awareness of the 2200 feet altitude of the FAF that we briefed for altimeter comparison. I believe the compression of altitude and heading changes lead to my loss of SA (Situational Awareness).Cause: In hindsight; my situational awareness of the FAF at 2200 feet should have caused me to question and confirm my read back to ATC of 2000 feet. The 2200 feet altitude was briefed for altimeter comparison during the approach brief. We followed all trained and standardized procedures and should have trapped the incorrect altitude internally without any reliance on ATC to correct a read back. We should have ask for vectors to set back up for the approach to allow us time to breathe through the event and set back up for the approach ensure ensuring we were okay.