Praise be to Allah.
It is not permissible for a woman to travel without a mahram, whether the journey is for an act of worship such as Hajj or visiting and honouring her parents, or it is a permissible journey such as a vacation and the like. The evidence for that is as follows:
1 – The Prophet (peace and blessings of Allaah be upon him) said: “No woman should travel except with a mahram, and no man should enter upon a woman unless her mahram is with her.” A man said, O Messenger of Allaah, I want to go out with such-and-such an army and my wife wants to go to Hajj. He said, “Go (to Hajj) with her.” Narrated by al-Bukhaari (1862).
Muslim (1339) narrated from Abu Hurayrah that the Prophet (peace and blessings of Allaah be upon him) said: “It is not permissible for a woman who believes in Allaah and the Last Day to travel the distance of one day, except with a mahram.”
And there are many ahaadeeth which forbid a woman to travel without a mahram; they are general in meaning and apply to all kinds of travel.
2 – Travel may be exhausting and difficult, and because of her weakness a woman needs someone to help her and stand by her. Something may happen that makes her lose her mind and behave contrary to her nature if she has no mahram with her. This is something that is often seen nowadays because there are so many accidents with cars and other means of transportation.
Moreover, travelling alone may expose her to temptation and evil, especially as there is a great deal of corruption. There may be people sitting near her who do not fear Allaah, and they may try to tempt her to do haraam things. So it is only wise that she should be accompanied by a mahram when travelling, because the purpose of having a mahram with her is to protect her and look after her. Travel is a situation in which unexpected things are expected to happen regardless of the amount of time the journey may take.
Al-Nawawi (may Allaah have mercy on him) said: The point is that everything that is called travel is forbidden for a woman without a husband or mahram. End quote.
More than one of the scholars narrated that the fuqaha’ were unanimously agreed that a woman is not allowed to travel without a mahram, except in exceptional cases.
Al-Haafiz Ibn Hajar (may Allaah have mercy on him) said: al-Baghawi said: They did not differ concerning the fact that a woman may not travel for anything but the obligatory Hajj except with a husband or mahram, except a kaafir woman who becomes Muslim in daar al-harb or a female captive who escapes. Others added: or a woman who becomes separated from her group and is found by a trustworthy man, in which case it is permissible for him to accompany her until he brings her back to her group. End quote from Fath al-Baari (4/76).
The scholars differed as to whether it is permissible for a woman to travel for the obligatory Hajj without a mahram. The correct scholarly view is that it is not permissible. This has been explained in the answer to question no. 34380.
What you have mentioned about taking the wife to the airport and her brother meeting her in the other country is not acceptable, rather the husband or a mahram must accompany her throughout the journey.
The wife has to obey her husband, especially if he tells her to do that which constitutes obedience to Allaah and he forbids her to do that which constitutes disobedience to Allaah.
You have to explain the shar’i ruling to her, and tell her that the believer has no choice when it comes to the ruling of Allaah or His Messenger.
Allaah says (interpretation of the meaning):
“It is not for a believer, man or woman, when Allaah and His Messenger have decreed a matter that they should have any option in their decision. And whoever disobeys Allaah and His Messenger, he has indeed strayed into a plain error”
“The only saying of the faithful believers, when they are called to Allaah (His Words, the Qur’aan) and His Messenger (صلى الله عليه وسلم), to judge between them, is that they say: ‘We hear and we obey.’ And such are the successful (who will live forever in Paradise)”
That should be done in a kind and gentle manner, not a harsh and cruel manner.
We ask Allaah to guide you both.
And Allaah knows best.