Is the grandmother sinful for selling part of the trust (gold) without the permission of her mentally ill sister to cover the expenses of her sister's treatment?

Your grandmother does not have the right to claim back from her sister what she spent on her, except under two conditions: that she intended to claim it back at the time of spending, and that this intention is proven by legitimate evidence.

If these two conditions are not met, then the gold taken by the sister is considered a debt upon her, to be repaid from her inheritance, and what the grandmother spent is considered a favor.

However, if both conditions are met, then the grandmother has the right to claim back what she spent, but this must be done through a Sharia judge, and the value of the gold should be deducted from the amount claimed.

It is necessary to seek forgiveness for the grandmother and strive to ascertain the truth.