Is paying a sum of money "under the table" to complete the purchase of an apartment—a sum that is not part of its actual price and is not mentioned in the contract—and then backing out of the purchase before its completion, considered falling under the ruling of a briber? And does this act have any relation to the Quranic verses?

Bribery is that which is given to invalidate a right or validate a falsehood, and it is one of the major sins. The Prophet, peace and blessings be upon him, cursed both the briber and the bribed. As for brokerage, it is permissible, and there is no harm in the broker taking a commission from the seller without the buyer's knowledge, or vice-versa. The amount of the fee is determined by agreement. If the amount requested by the broker is in exchange for deceiving and misleading the owner, then this is impermissible and constitutes forbidden deception. You did well to retract from this sale.