Air carrier aircraft Captain noticed a light aircraft about 600 feet from them on approach to a different runway. Reporter stated they did not hear any radio transmissions from the light aircraft on the common frequency.

Narrative

Inbound on RNAV (GPS) Runway 18 approach to IRK. Captain was PF. Weather good VMC. Making position calls on CTAF.Noted traffic call on CTAF while passing DELOO inbound about 10 miles from touchdown. Aircraft Y turning final approach for Runway 18. We were unable to see their position on TCAS; or visually; but continued inbound while slowing to final approach speed and continuing to broadcast our position.At about WOGED; 5 miles from RWY 18; on stabilized approach in final landing configuration at about 140 KIAS; we heard an unidentified call; 'there is traffic on final approach for runway 18.' We considered abandoning the approach at that time; but not certain of the exact location of any conflicting traffic we were concerned abandoning our stabilized approach could make any potential conflict possibly worse.At about 3 miles from touchdown; and CA 1;000 ft AGL; I was able to visually confirm a similar High Wing Aircraft taxiing into the FBO parking ramp. Believing this to be the conflicting traffic; we continued our stabilized visual approach.At that point; my FO was startled to see Aircraft Y beneath us and off to our right side; apparently in a low pattern for the turf Runway 9-27. Our best estimate was about 600 feet minimum slant-distance from us. At that point; he apparently spotted us and broke off his maneuvering to the west. Seeing Aircraft Y was safely clear of us; we elected to continue our stabilized approach to a normal landing.After parking and securing our aircraft at the terminal; we saw Aircraft Y taxi into and park at an FBO. We approached the pilot and his student to discuss the circumstances of our traffic conflict.The pilot was a local CFI with a student receiving their tail wheel endorsement. He briefed his aircraft had no electrical system with no onboard radios or transponder. He claimed to be using a handheld radio; but we never heard any transmission from him; nor did he recall receiving any from us; despite our multiple position calls on CTAF while inbound.The airport manager subsequently contacted us for details on our traffic conflict event and indicated he would speak with the CFI about the need for better communications on CTAF when in the pattern at IRK and its locally published notice of frequent; high-speed; scheduled airline traffic.