Hayden has made 171 appearances for the Premier League side since moving to St James' Park in 2016 although has not played for them since 2021.
The 28-year-old spent the first half of this campaign with Belgian side Standard Liege.
Hodge, 21, has made six Premier League appearances for Wolves so far.
The Republic of Ireland Under-21 international started out at Manchester City before moving to Molineux in June 2021 and is the second player to join QPR from Belgium this week, following the arrival of striker Michael Frey from Royal Antwerp.
"I am very happy we have been able to bring in Joe. He is a player who is very comfortable in possession, very good in small spaces, very good in the passing game," R's boss Marti Cifuentes said.
Hayden added: "It is obvious to see the position the club are in at the moment but it is a challenge that I feel I can add some value to, on and off the pitch - be an experienced player and be a leader to try and help the club stay in the league."
QPR are currently third from bottom of the Championship, three points from safety ahead of Saturday's trip to Blackburn.
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