So, I’m not particularly aware if Vargas and Perón had a personal relationship or not. Many similarities have been drawn between the two because of their similar forms of populist nationalism. However, that does not mean that the two were allied in a political sense, or friendly in a personal sense. I have my doubts as to whether they were friendly with one another because of their respective countries’ position in South America, especially in relation to the United States. Because Vargas was particularly supportive of the United States throughout the 1930s and 1940s, his government received all the aid and support that the United States could give in Franklin Roosevelt’s efforts at continental solidarity against possible fascist penetration and subversion in Latin America. It also helped that the Brazilian economy and the American economy during this time were very compatible, with Brazil providing raw goods like coffee and sugar which the United States was incapable of producing due to climate.

Argentina on the other hand, especially during Juan Perón’s presidency, sustained quite a tense relationship with the United States. The military government that was in power for three years before Juan Perón’s election was very hesitant to throw their hat in with the United States during the Second World War, partly because of military sectors’ sympathy for Italy and Germany, but also in accordance with Argentina‘s foreign policy adopted during the previous world war, that being strict, unyielding neutrality. Observing what was happening in Brazil and other countries in the region, Perón as president was very hesitant to ally Argentina with the United States, instead developing what was known as “la tercera posición”, or the Third Way in between the unbridled capitalism of the West and the collectivist communism of the East. Therefore it is very likely that Perón viewed Brazil, whether under Vargas or otherwise, not as an ally but as a rival in becoming South America’s political and economic leader.

Yet again, the fact that I’ve never read anything about if the two leaders had a personal relationship doesn’t mean that it didn’t exist. It wouldn’t be the most shocking thing to learn that that’s the two were friendly with one another. I’m not entirely sure what book you read where the author indicated Perón’s support for Vargas in the 1950s, but I would love to check it out if you remember the title.

EDIT: Just a little addendum, I have my doubts about the credibility of the book you mentioned, since Vargas didn’t run for president in 1954. He was reelected in 1950 after being out of office for five years, but then shot himself when he heard that a coup d’état was in the making. But also, I don’t study Brazil specifically, I study Argentina. So I may be wrong about this.