The first sentence is correct, but it would be even better to say: Du kannst dich mir ruhig anvertrauen. The reason for the order is easy: Both pronouns must come immediately after the conjugated part of the verb, and the order with 2 pronouns is always first Akk., then Dat.

There are 3 kind of orders: 1) With two nouns, Dat. comes first, then Akk., e.g. ich beantworte dem Mann die Frage. 2) With one pronoun and one noun, the pronoun comes first, it doesn't matter if Dat. or Akk., ich beantworte sie dem Mann, ich beantworte ihm die Frage. 3) With two pronouns, the Akk. comes first, e.g. ich beantworte sie ihm. With 'anvertrauen' it would be exactly the same.