Recently, my aunt told me something which her daughter had said, and that made me doubt her daughter's faith. What I can say is that it sounded as if she has no trust in Allah. My aunt told me not to discuss it with people. Am I allowed to discuss what her daughter said with people?

All perfect praise be to Allah, the Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad, sallallahu ‘alayhi wa sallam, is His slave and Messenger.

It is impermissible to discuss this matter with people. You did not hear it from her directly; rather, it was transmitted to you through your aunt. The basic principle is that this girl is innocent of what was attributed to her until it is proven that she actually said it. If it is proven that she said it, then the one who heard it directly from her should give her advice wisely and kindly. Tameem Ad-Daari, may Allah be pleased with him, narrated that the Prophet, sallallahu ‘alayhi wa sallam, said, "Religion is (sincere) advice." They said, "To whom?" He replied, "To Allah, His Book, His Messenger, the leaders of the Muslims, and their masses." [Muslim]

You are safe as long as you have not heard these words directly from her, so do not investigate the matter any further. The Prophet, sallallahu ‘alayhi wa sallam, said, "An aspect of the perfection of one’s Islam is to leave that which does not concern him." [Ahmad and At-Tirmithi] Moreover, sharing this information with people is of no benefit.

Allah knows best.