So, apparently Modern Western Square Dancing has a cousin called “Quadrilha” in Brazil.
The quadrilha (square dance) originated in Holland and was introduced in Brazil during the Regency period. It was very popular at the 19th century balls of the Brazilian elite, especially in Rio de Janeiro, where the royal court was located. Later, it descended the palace steps and gained popularity among the general public, who added new steps and changed the music.
Today, the quadrilha is a tradition at the June Festivals. As the name says, these lively celebrations occur in June, but often extend into the following months. The events are led by an announcer who calls out the dance steps.
The dancers, usually couples, dress in peasant outfits with straw hats and checkered shirts. The traditional quadrilha dance represents a wedding party; the “bride and groom” open the dance, followed by their “guests”, with a lively dance that includes many different steps and moves.
The quadrilha is accompanied by an accordion, triangle and drum, as well as a 4-string and 6-string guitar.
Source: Quadrilha (Square Dance)
It’s a bit different from Modern Western Square Dancing, but you can clearly see that they come from the same roots. Here is a video that shows one quadrilha dance being performed for an audience. (That is one major difference between the two. MWSD is rarely performed for an audience.)
Square Your Dance 
This isn’t the only square dance connection to Brazil. Henry Ford’s love of square dancing (primarily New England Quadrilles; his book Good Morning! inspired Lloyd Shaw’s publication of Cowboy Dances, which, in turn popularized “traditional” western squares and, eventually, MWSD) is well known, and lives on in the form of Lovett Hall, the Dearborn dance hall that he built for his favored dance master, Benjamin Lovett. What’s not as well known is that in order to reduce his dependency on British-controlled rubber from southeast Asian plantations, Ford attempted to establish his own rubber plantations in Brazil’s Amazon River basin. The company town for the plantation, creatively named Fordlandia, featured a dance hall, and square dancing was required.
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