OTTAWA — A hit-and-run car crash that injured two members of the Canadian Armed Forces in a small Quebec city on Monday morning quickly prompted widespread speculation that the motive might have been terrorism-related.

A sedan struck the two people, one of them apparently in uniform, as they walked in Saint-Jean-sur-Richelieu, a city south of Montreal that is home to several military facilities.

The local police chased the car for a few miles before it veered into a ditch and overturned. The driver, the police said, climbed out and confronted officers before being shot.

The 25-year-old man, whom the police did not identify, died later. The members of the military were also not identified by the police. One was in critical condition on Monday evening, according to the provincial police, who took over the investigation. The other military member had less severe injuries, the police said.

An unusual question in the House of Commons, which arose shortly after the crash, raised suspicions that what at first appeared to be a violent assault involving an automobile might have been an act of terrorism.

Randy Hoback, a member of the governing Conservative Party who represents a district in Saskatchewan, asked Prime Minister Stephen Harper about "unconfirmed reports of a possible terror attack against two members of the Canadian Armed Forces."

Mr. Harper, referring to notes, replied that "we are aware of these reports, and they are obviously extremely troubling."

Tom Mulcair, the leader of the opposition New Democratic Party, swiftly criticized Mr. Harper for being too quick to suggest motives for the crash.

"I was a bit surprised by the interplay between the prime minister and one of his backbenchers when we're still waiting for any information from the police," Mr. Mulcair said. "Let the police do their job, and then we'll know whether we're dealing with the type of situation they've described."

Early on Monday evening, however, Mr. Harper further fueled speculation with a statement from his office.

According to the statement, the motorist was known to a special antiterrorism program led by the Royal Canadian Mounted Police. "Federal authorities have confirmed that there are clear indications that the individual had become radicalized," the statement added.

While the statement did not describe the hit-and-run crash as an act of terrorism, it did say that the prime minister had been briefed on the episode by his national security adviser, the head of the Mounted Police and the country's top military commander. The statement provided no other details, and Mr. Harper's office declined to comment further.

The Sûreté du Québec, the provincial police force, offered little information about the dead man or his possible motives.

"The theory that this was a deliberate act is part of what we're looking at," Lt. Guy Lapointe, a spokesman, told reporters. "It's really too early to speculate."

The violence occurred after Parliament had approved sending jet fighters to join in airstrikes in Iraq against militants from the Islamic State.

Like their counterparts in other Western nations, security officials in Canada have publicly raised concerns over the possibility that Canadians would become radicalized and either travel overseas to join terrorist groups or commit terrorist acts at home.

At a House of Commons committee hearing last week, Michel Coulombe, director of the Canadian Security Intelligence Service, estimated that 130 Canadians had left the country to join what Canada defines as terrorist groups.