Airport spokeswoman Heather Hamilton confirmed a Bombardier Q400 had experienced a problem after landing on the runway.
The Air Canada Flight 8481 left Calgary with 71 passengers and four crew members on board.
While the plane was in the air, the pilot realized that a tire had blown. Strong crosswinds in Calgary meant the plane had to be diverted to Edmonton.
Something went wrong during the landing, Hamilton said.
"It landed in Edmonton but then there was some incident shortly after it landed," she said. "And as a result of that, there were injuries to three passengers."
Hamilton said a fourth person accompanied one of the passengers to hospital. According to Jazz Aviation LP, the company that operates as Air Canada Express, one person remains in hospital.
Transportation Safety Board investigators arrived on the scene Thursday night and will return Friday morning.
Once the Transportation Safety Board investigation concludes, airport authorities and Air Canada will remove the plane from the runway.
After the plane is removed, the airport will inspect the runway. It will remain closed until midnight, but could reopen earlier. No flights have been delayed as a result.