Is it permissible for a sister to pay a bribe to forge an academic certificate with the approval of her husband, who studies Islamic jurisprudence, and is he who opposes that considered mistaken in his prohibition of cheating?

Falsifying certificates is forbidden (haram) because it involves lying and deception. The sin is compounded by paying a bribe for it. Allah Almighty has warned against speaking falsehood (az-zur), and the Prophet, peace be upon him, considered it among the major sins. The Prophet also cursed both the briber and the bribed. It was your duty to advise your sister and clarify the error of what she intended to do, not to quarrel and curse, for cursing a Muslim is an act of transgression (fusuq).