Is it legislated to go to the girl's father and speak with him to find out the real reason for the refusal after her mother convinced her with illogical reasons, or is what is happening a sign that the matter will not be facilitated, knowing that her family are Christians?

If the girl's family are not Muslims, they have no guardianship over her, because a non-Muslim cannot have guardianship over a Muslim. The Muslim ruler would be her guardian. If there is no Muslim ruler, she may appoint a just Muslim man to solemnize her marriage contract, or a just man may marry her off with her permission. The girl must be informed of this legal ruling gently and kindly. It is preferable that this be done through a woman from the questioner's relatives. If she agrees, then all is well. If she insists [otherwise], then one should turn away from her. This is while emphasizing the impermissibility of relations between non-mahram men and women.