Air Traffic Controller reported a situation where there is an airborne conflict and the ATC automation doesn't issue a conflict alert due to another facility suppressing the conflict alert function resulting in a NMAC.

Narrative

I was the R-side working XX and XY combined with a D-side training team helping out. We were in the middle of a departure push with a few extra things going on that added to complexity such as multiple aircraft on approach; aircraft handed off to us on headings that were not coordinated; and multiple overflights impacting our RNAV departures out of D10. It was just a bit busier than normal. Aircraft X was performing a practice approach at TRL to Runway 35 and was on the left downwind when I noticed there was a mode C intruder target that was eastbound just one hundred feet below them on course to merge with them in less than one minute. The conflict alert had not activated. I called traffic to Aircraft X at XA:01 and the aircraft did not respond. I immediately followed this up with a traffic alert and turned the aircraft 90 degrees to the right to miss the traffic. When the radar updated as I issued the alert; the aircraft were at the same altitude on course to merge in less than 30 seconds and both aircraft had mode C returns of 3;000 feet. The conflict alert never activated and Aircraft X eventually got the other aircraft in sight. This is the 2nd report I have submitted for this exact issue and the 3rd time I've seen it create an unsafe situation on position. These two aircraft were going to collide and we had no warning of any conflict because the conflict alert has been suppressed inside of D10's airspace. This seriously needs to be looked at because several unsafe situations have resulted from this and someone easily could have missed this traffic confliction because we were never warned of the conflict. Sector traffic and complexity were both high; making it unlikely the controller would have seen the conflict.