In English, we have a phrase that goes “Cross my heart, hope to die(, stick a needle in my eye)” which acts as a way to convey trust. In Brazil, it’s not so literal but there is a comparable phrase. Before I get to it, another English phrase is “I swear on my (insert relative here)’s grave” which means the same. In Brazil, one swears on the health of someone important to them.
Ex. I swear! Cross my heart, hope to die.
Ex. Eu juro! Pela saúde da minha mãe.
Of course, it doesn’t have to be one’s mother. You can also say “juro por Deus” (like “I swear to God”).