Meaning
To talk nonsense / To talk rubbish
Example
Não leve a sério o que ele disse, ele estava bêbado e falando abobrinha.
Don't take what he said seriously, he was drunk and talking nonsense.
Origin
Here abobrinha ("little squash/zucchini") was the nickname of an old Brazilian banknote: the 1,000-cruzeiro note that circulated from the 1940s to the 1960s, called that because its colour recalled the squash fruit. As galloping inflation ate away its value, the once-prized note became practically worthless.
Sources: HR Idiomas, Portal Litoral Sul.
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More idioms
Acertar na moscaTo guess correctly / To hit the nail on the headAmigo da onçaA fake/false friendAndar na linhaTo toe the line / To behave properlyArmar um barracoTo cause a public sceneRodar a baianaTo cause a public scene / To go ballisticArrumar sarna pra se coçarTo ask for troubleArrastar as asasTo hit on someone / To flirtAbrir o jogoTo come clean / To open upArregaçar as mangasTo roll up one's sleeves / To get to workAgarrar com unhas e dentesTo hold on for dear life / To seize an opportunity