No, a woman didn't ride in a hearse smoking a pipe in Quebec City.

The story of a live woman in a hearse riding around and smoking a pipe in 19th century Quebec City that's been circulating on the Internet for a few years likely never happened. The sole source for it is from The Illustrated Police News, a sensational Victorian newspaper published in England that reads like the Weekly World News - fiction and fact mixed together. The short article gives no source for their story, no date and no name for the woman.

The Société historique de Québec (Quebec City Historical Society) called it "Fake News" in a Facebook post  after having searched all Quebec newspapers from several years of that period and finding nothing about it. The mystery woman's actions, as well as the owner of the hearse, would have attracted a lot of attention, and probably legal action. The street they were supposed to have driven on leads to the Cathedral of Notre-Dame, the oldest church in Canada, and they could easily have faced criminal charges. Professor Brigitte Garneau, who has written several books on the funeral industry in Quebec in the 19th Century, said she has never heard of it.

The churches were powerful in Canada - in the 1880s, at the request of the local curate, two shopkeepers were charged in Montreal after putting reproductions on display of statues by Michaelangelo, Night and Day, two nudes which are on the tomb of Giuliano de Medici in Florence. Jehovah's Witnesses were prohibited from distributing tracts in Quebec City (see Saumur v. Quebec City) in the 1940s and even as late as the 1950s Elvis was banned from performing in Montreal at the insistence of the church. One of my teachers told me that in 1950s in Toronto that the police would show up to disperse kids playing ball on Sundays because that was against the law.

April 11, 1872, "The Illustrated Police News
 London, England 

"A Quebec Woman Creates a Sensation, Riding Through St. John Street in a Hearse, Reclining on the Coffin-Bed, and Smoking a Pipe.

What will women do next to distinguish themselves, we wonder! A female in Quebec, the other day, perpetrated a ghastly joke, mocking death in his own domain, by lying down in a hearse and smoking a pipe in a funeral chariot was driven through the street.

If this exhibition had been made in the United States, our neighbours at the North would have made it the subject of very strong animadversions."

The "What will women do next..." is obviously refering to the growing women's suffrage movement, aimed at getting the vote for women in the UK. Quebec women were the last in Canada to get the vote, in 1940.

I'm nor sure what the last line refers to - perhaps it's a joke about some event that was current at the time.