In January 21, 1948 a new flag of Quebec was officially adopted in Canada by Maurice Duplessis, 16th Premier of the Canadian province of Quebec. Today its floating over the main tower and many other government buildings representing French Canadian heritage.
The official name of the flag is “Fleurdelisé”. Cross on flags is very common in Europe (England, Denmark, Finland, Greece, Iceland, Malta, Norway, Scotland, Switzerland, Sweden and many other examples), this is the main symbol of Christianity. “fleurs-de-lis“, a small white French symbol on blue background means purity and royalty, and the blue background represents heaven.
It was an earlier version that showed gilt fleurs de lis all pointing inward to the intersection of the arms of the cross. The final version, with white fleurs de lis pointing upward, apparently had not yet been available.montrealgazette.com
You can watch a video by J.J. McCullough with a short yet informative history of Canadian flags, including the flag of Quebec (4:40):