Cartier didn't go to Brazil, that was another, far more obscure expedition which failed IOTL.
The idea of colonizing this region comes primarily from Nicolas Durand de Villegaignon, born in Provins in 1510, gentleman and soldier, knight of Malta and vice-admiral of Brittany, whose classmate was Calvin at the University of Orleans. In 1553, he pleaded with Gaspard de Chatillon-Coligny, admiral of France, for a project to colonize Brazil. The latter quickly understood the political interest of such a colonial expansion that would consist in dealing a blow to Iberian hegemony in America, which easily convinced Henry II: “A young and strong France would emerge overseas". Basically the idea of colonizing Brazil became a popular trend among the naval nobility and traders.
I find lack of Jacque Cartier strange. If French can conquer Brazil from Portuguese, surely they can do the same to Protuguese Canada.
They will probably focus aggression on the viceroyalty of the Plate River. In fact I would say that Portugal and France will have very bad relations in this regard. Portugal does not have the strength to hold the plata basin. So in the long term, unless there is a debt intervention (aka the biggest luck of the century) this is a complicated region. With the colonies not being very productive, Portugal will focus heavily on Asia (much more than in OTL). For example, the dispute over Brazil between Portugal and France. Had a intense French maritime activity is accompanied by numerous acts of piracy and aggression against the Portuguese to try to expel them from the region, both in Brazilian waters (the first French ship was boarded in Brazil in 1504, in the bay of the river Paraguaçu), as well as at mandatory crossing points on return routes (Azores, Ireland Sea and other important connections). The situation got so bad that the French Atlantic merchant and military navy was attacking any Portuguese ship they encountered on the way to Brazil. Basically this navy (France had two for each ocean, the Atlantic and Mediterranean) was in an open war with the Portuguese navy (even if France was not) That is why, in a spirit of appeasement, the French and Portuguese sovereigns, after signing a treaty of friendship in Lyon (July 14, 1536), they convened in Bayonne, in 1537, a Court of Awards, intended to resolve conflicts between French and Portuguese shipowners. This Court, however, did not bring any agreement, except that Francisco I undertook to prohibit his subjects from trading in Brazil. The Atlantic navy obey (kinda, becausee it continued to act aggressively towards the Portuguese).
I wonder why Francis I sent the expedition to Brazil instead of Canada.
Well about Brazil and France. (it's kind of a reused and modified answer from another thearth about this):
The king was not the main figure in the colonization of this region. The French Atlantic Navy was the group pushing colonization (or rather the admiral and the cities with influence in the Atlantic Navy ). These naval and commercial elites, the Admiralty of Rouen (atlantic navy) and the merchants/nobility of the city as a whole and set up an immense decoration representing the Brazilian lands with their forest, cutting and transport. burning wood, animals and “natural” fruits for the king of France. This is partially due to the dispute between the Atlantic navy and the Mediterranean navy. The Atlantic navy was very decentralized and this colony and (future ones) would provide the funds needed to reform and truly centralize that navy. Brazil curiously had an important place in the French imagination, Brazil has been for France the land of all commerce: ""commerce of souls" for Protestants or Catholics ( from Fort Coligny, in Rio de Janeiro, to Fort Saint Louis, in Maranhão), "product trade" by smugglers, pirates or privateers (in Paraíba, São Vicente or Rio de Janeiro for Dugay Trouin, for example) and " trade of ideas” by scientists and artists (from La Condamine to the French Artistic Mission of 1816).
In OTL the concern for France was so great that the Portuguese king, at the beginning of the 16th century, published a decree ordering all his subjects, under penalty of death, to sink all French ships leaving or arriving from Brazil. And that was not without reason, in 1540 nine ships from Rouen were send to Brazil; fifteen Dieppe ships and ships from Brittany are equipped for the same destination (24 in total). In 1541, thirty to forty ships left for South America and especially Brazil. In 1546, a fleet of twenty-eight ships left Le Havre bound for Brazil.
Parmentier complained that the King of France (did not take seriously the influence of the French among the inhabitants of Brazil: "If the King of France had wanted to free himself a little from the French traders in less than four or five years, they would have won the friendship and assured the obedience of the people of these new lands, and this without any other weapons than persuasion and procedures”. For the great captain, in the space of these four or five years “the French would have penetrated further into the interior of the country than the Portuguese in fifty years , and the inhabitants would probably persecute the latter” * - Barbosa, Mario de Lima, 1923,
Les Français dans l’histoire du Brésil, Rio de Janeiro-Paris, Briguiet & Blanchard.
The King in OTL was more of a hindrance than a help.