Is it permissible to modify the eyebrows after having vowed to leave them as they are in exchange for conceiving a child, and what is the expiation for breaking this vow, and will this negatively affect the son?

Thinning the eyebrows by plucking is forbidden (haram). As for trimming them by cutting or shaving, there is a difference of opinion, and it is better to avoid it unless the hair is excessively long to the point of disfigurement, in which case it is permissible to remove some of it by cutting or shaving, but not plucking. If the questioner is in a situation where it is permissible for her to trim her eyebrows, then her vow is not binding according to many jurists. If she is not in such a situation, then her vow is also not binding, and she must leave them as they are. A vow does not alter Allah's decree, and piety (taqwa) is a reason for the continuation of blessings.