Quincey, 31, signed a one-year, $1.25 million contract with the Devils on Sept. 28, according to CapFriendly.com. He can become an unrestricted free agent July 1.
"I'm very excited to help whoever it is with the little details that make a difference," Quincey said. "Those little things go a long way, especially in a playoff run. One little shift can make or break a series."
Quincey has 12 points (four goals, eight assists) in 53 games this season.
"Kyle did a great job and played a key role for us on defense," Devils forward Kyle Palmieri said. "I played against him a ton when he was in Detroit and he's always been a good, consistent, player."
Quincey has 152 points (34 goals, 118 assists) in 548 regular-season games in 12 seasons with the Devils, Detroit Red Wings, Colorado Avalanche and Los Angeles Kings. He has eight assists in 54 Stanley Cup Playoff games.
Prout, 26, is in the first year of a two-year, $3.15 million contract with an average annual value of $1.575 million. He can become an unrestricted free agent on July 1, 2018.
"There were a couple ways to go to assess the market in terms of draft-pick value, and knowing we already had 10 (draft) picks, the Quincey for Prout [trade] made sense for us," Devils general manager Ray Shero said. "We like Dalton Prout as a player … he's got a year left on his contract, which was attractive to us, and brings a physicality and hardness and he's going to have some time to get his game back here."
He has three assists in 15 games for the Blue Jackets this season. Prout has 33 points (six goals, 27 assists) in 224 NHL regular-season games in six seasons, all with Columbus.