What is the expiation for a sister telling her brother about her suspicions regarding a friend of hers, and what should she do with her brother after he became suspicious of her?

If you spoke about your friend out of fear and concern for her, intending to save her, then there is no blame upon you, and no expiation is required of you. However, you should have paused. If you indulged in backbiting her, then you must seek her forgiveness. If what you said reached her, apologize to her and ask for her pardon. If it did not reach her, do not tell her; instead, frequently supplicate for her, seek forgiveness for her, and praise her in front of those with whom you spoke. As for your brother's suspicion, its cure is through prayer for him, your adherence to hijab, lowering your gaze, avoiding non-mahram men, increasing in voluntary acts of worship, being frank with him, and warning him against ill-suspicion. If he sees your steadfastness, his suspicion will vanish.