Probably Quinn only due to high Democratic turnout in Cook County.
Turnout isn't high in Cook it's just highly Dem. Brady lost to Quinn because he couldn't cut into enough of the Cook/Chicago result. In Chicago Brady got 17.5% of the vote, but Kirk got 19.5% on the way to his statewide win. The conventional wisdom is that if a Pub breaks 20% in Chicago they will win statewide.
Half of Cook is in the suburbs, but include some very strong black and Latino areas. Brady lost suburban Cook by over 100K votes but Kirk lost by just over 60K. Kirk is from the suburbs and heavily campaigned there, compared to Brady who was significantly less visible.
Rauner is devoting most of his effort to Chicagoland to try to replicate Kirk's turnout pattern.