Is it permissible for a married woman who is suffering from her husband (who is unemployed, prays and backslides, and curses God) to complain to her cousin’s son, who previously loved her and proposed to her, knowing that she fears God and has children?

You mentioned that your friend’s husband curses Allah, and this act is considered an apostasy from Islam by the consensus of scholars. Whoever curses Allah, the Most High, has committed an act of disbelief, whether in jest or in seriousness. If his disbelief is established, then it is not permissible for him to be married to your friend, as the marriage of a disbeliever to a Muslim woman is unanimously forbidden. Therefore, your friend must rid herself of this man, as her continued relations with him are considered fornication (zina). She must also stop communicating with her cousin’s son and beware of Satan’s enticement to speak with him, especially regarding matters of love.