Is it permissible for me to buy gold from my brother in installments, knowing that it is an inheritance from our mother, and it has emotional value, and the price will be paid and the gold received later at the same purchase price?

If the agreement between you was merely a promise to purchase, then there is no harm in it. However, if it was a contract of sale and purchase, it is invalid according to the majority of scholars due to the absence of immediate possession (taqabud) at the contract session, which is a condition for selling gold for currency. This is based on the general statement of the Prophet (peace be upon him): "Do not sell what is absent for what is present from it." And banknotes take the ruling of gold in terms of their monetary value. The way to resolve the difference of opinion is to postpone the contract for purchasing gold until the meeting, when the price can be handed over and the gold received.