The 31st of March is a relevant date as at 3:00 in the morning in 1964, General Olímpio Mourão Filho mobilized the 4th Division and moved towards Rio de Janeiro, setting in motion the military-civilian plot to overthrow President João Goulart (a.k.a. Jango). The plans were for the 4th of April but Mourão Filho was hotheaded and, once his action was in progress, could not be reversed. But it would have been funny if he had made this on the 1st of April.
The military at the time was as confused as the rest of the country and divided on loyalty and ideology, thus splitting into rebel and legalist camps. Before midnight legalists in Rio de Janeiro state were already moving towards Mourão Filho's column. Note that Rio de Janeiro was no longer the capital but still very important as Brasília was very new. In the days that followed units from both camps moved to seize key positions, almost came into conflict and there was a specter of civil war, but ultimately the legalists chickened out and were defeated without any shots fired. Jango's overthrow happened in both military and civilian fields, and in the latter legalists also folded. In a handful of days Jango had fled to Uruguay and Congress -the same Congress, it was not dissolved- began the transfer of power.
I'm trying to make sense of the allegiance of specific units, their movements and the importance of the ground they sought. However I can't even find an order of battle for the whole military at the time, it seems such a thing only exists for the 21st century. For figuring out an order of battle it's easier to read the historical material published by the present successors of old units. The historical record registers important units through the crisis but not the whole hierarchy. Additionally information on some states and the units in them is only found when specifically searching for that state, with wider histories only covering Rio Grande do Sul, São Paulo, Minas Gerais and Rio de Janeiro.

From what I gather the higher level was of Military Commands/Armies -presently only the former is used, and probably only the latter then- with the most important being the Ist (eastern), II ("central", now southeastern) and IIIrd (southern). Below those were Military Regions/Infantry Divisions:
The Ist Army had the 1st Division in Rio de Janeiro and Espírito Santo, strong and with plenty of legalists, and Mourão Filho's 4th Division in Minas Gerais. In Rio de Janeiro cadets at the Agulhas Negras academy also played a role and count as a unit for this context.
The IInd Army had the 2nd Division in São Paulo. It took the rebel side.
The IIIrd Army had the 3rd Division in Rio Grande do Sul. It was a legalist bulwark. It also had the 5th Division in Paraná and Santa Catarina, but it was much weaker. I just found an article about it and it had a complex situation ultimately won by the rebels.

Among governors there were also legalists and rebels. They didn't necessarily share the allegiance of the military in their states: in Rio Grande do Sul Ildo Meneghetti transferred the state's administration to the hinterland to evade legalists in the capital, while Guanabara (a single-municipality state comprising Rio de Janeiro proper, a legacy of its capital status) also had an anti-Jango governor. Minas Gerais and São Paulo's governors also sided against the national government. Rebels arrested governors in Pernambuco and Sergipe.
There's a lot of reading to do, I'll look into it place by place and try to draft a map of the situation with what I find.