Slovenia international Kelhar, 30, who won the Serbian title with Red Star Belgrade last season, had trained with Wednesday during pre-season and signed a short-term deal in time to be included in the squad for the opening-day win at Brighton.
"We sent him off to get a scan and he's not come back with a good report," Owls manager Stuart Gray told a Thursday press conference.
"He's done his anterior cruciate ligament so he's going to need major surgery.
"He got the injury in the Newcastle (pre-season) game and the specialist has said he's torn the ligament in his knee, which needs to be repaired.
"So I can't see the boy playing again for us this season unfortunately. It's the curse of the centre-halves which seems to have carried on from last season.
"It's a big blow for the player, he's a wonderful man, he's distraught about it.
"It was an innocuous challenge in the Newcastle game. He tried training with it and was on the bench at Brighton, but his knee has blown up so it's not good news for him and not good news for Sheffield Wednesday."
The blow has been softened by the recent return to contention of fellow central defender Kamil Zayatte, who missed the second half of last season following ankle surgery.