No motive has been disclosed, and police said they were working to determine a link between the gunman and the couple.
Police said Costello and Burns were responding to calls of shots fired at an apartment complex and saw two deceased civilians before being shot and killed themselves.
Fredericton police Chief Leanne Fitch said Raymond used a long gun and was in an elevated position when he fired. Fitch said he was shot by police and was in serious but stable condition.
Judith Aguilar, an office manager for Sunfield Apartment Rentals, said Raymond lived in the complex for about four months and was an avid cyclist who often came to pay his rent in cash while wearing a bike helmet.
“He seemed like a very normal and pleasant person, really,” she said. “He’s tall and was in fairly good shape because he biked everywhere.”
At one point, she said maintenance workers needed to do some work in his apartment and he didn’t want the workers there while he wasn’t present.
“He seemed concerned, he said he had an expensive computer and an expensive bike,” she said. “They didn’t even have to go all the way into the apartment, they were just fixing his doorframe at the time.”
Residents were stunned by the episode, which took place in a city of 60,000 that last saw a homicide in 2014.
But the shooting comes as Canada wrestles with a string of violence, including an instance in Toronto last month where a man with a handgun opened fire in a crowded part of the city, killing two people and wounding 13 before he died in the confrontation.