The Owls had the opportunity to do some damage in the 18th minute when Fernando Forestieri raced clean through on goal, only to slip at the crucial moment and see his chance go begging.
An injury to Modou Sougou forced almost 10 minutes of first-half stoppage time, during which Chery came within inches of firing QPR in front, but his effort flashed over and the sides went in goalless.
Early into the second half and the game still looked evenly matched, but Matty Phillips almost fired QPR ahead with a rasping strike that flew over in the 58th minute.
QPR staged another attack in the 66th minute, led by Chery. He picked out Phillips, whose initial shot was blocked before Tozser blazed a follow-up into the stand.
Both sides had late chances to win the match as Westwood foiled Chery from point-blank range at one end of the park, before Bannan blew a late opportunity for Wednesday by firing over unopposed at the other.
The result leaves the Owls one point better off than QPR in the table, lying in ninth place on 17 points.