The domestic flight from Calgary to Toronto was canceled "following an issue of non-compliance," and the family "refusing" to deplane, a WestJet spokeswoman said in a statement.
"Due to the rapid escalation of the situation on board, our crew felt uncomfortable to operate and the flight was subsequently canceled," the spokeswoman added.
The family in question - 31-year-old Canadian Safwan Choudhry, his wife Marium Choudhry, and their three-year-old and 19-month-old daughters -- were traveling home to Toronto after visiting relatives in Calgary, Safwan Choudhry said in an interview with CNN.
Choudhry told CNN that the incident unfolded very quickly and "was the most embarrassing thing we have gone through."
Shortly before takeoff, he told CNN, a crew member approached his wife and asked that both children wear a mask -- even though, he said, Canadian transportation department Transport Canada's policy does not require children under two to wear one.
WestJet denies that it asked the younger child to wear a mask.
In a statement issued to CNN in response to the family's allegations, the spokeswoman said: "Transport Canada's interim order does not require infants under the age of two to wear a mask but does require any child over the age of two to wear one. It is important to clarify that WestJet did not request the infant under the age of two to wear a mask."