Is it permissible to place a listening device in the husband's car to find out what transpires between him and a woman with whom he travels alone, given that the husband denies it and disregards the sacredness of the matter?

It is not permissible to place a listening device on the husband, because it is a form of spying, which is forbidden by Islamic law, and what is mentioned in the question does not justify it. The husband should be reminded of the impermissibility of a man being alone with a non-mahram woman, and that Satan is the third of them.