Air carrier Captain reported an NMAC during visual approach to BHM requiring the crew to adjust the descent rate to avoid a possible collision.

Narrative

While descending into BHM; Birmingham Approach called out traffic at our 10-11 o'clock that was VFR at 4;500 feet and converging. Neither the FO (First Officer) or I saw the traffic and we told ATC this. We were descending down to 5;000 feet while on vectors for a left downwind for a visual approach to Runway 06 at BHM. As we came through 6;000 feet; we received an amber 'TRAFFIC' message and associated audio cue. As I was reaching up to slow my rate of descent from 1.5 to 1.0; we received a 'MONITOR DESCENT RATE' audio cue. At almost the same instant; the autopilot had switched to ALTS CAP causing the aircraft to go into PITCH Mode because I had started adjusting the descent rate with the wheel. We leveled off at 5;000 feet and it was at about this time my FO and I both saw the previously called traffic pass 500 feet below us.Suggestion: Shallow descent rate sooner when traffic is known to be in the area and might be a factor. We did not see the traffic because of ambient lights from the city of Birmingham.