Is it considered forgery to state "the deferred dowry has been paid" in the marriage contract at the court when it has not been paid, and should this deferred dowry be paid to the wife's father or should a complaint be filed?

What is relied upon in religious matters is the verbal agreement, not the written document. So, if the agreement was to pay the deferred dower at a specific time, you are not obligated to pay it before its due date. If there is no specific due date, then it becomes due upon separation (divorce or death).

As for the written document created after the contract, if there is an error in it stating that the deferred dower has been paid, this is not forbidden deception, but rather an error that should be corrected to reflect the reality. If it is possible to amend the document to match the agreement, that is good. If it is not possible and the wife wishes to secure her right, then either you pay her the deferred dower, or you provide her with a document proving her right to it.