What is the ruling regarding a wife studying for a master's degree in English in another country, on a scholarship from the state, with her husband accompanying her, and the possibility of addressing infertility issues during this trip?

If the studies are free from unlawful matters, and residing in that country does not cause temptation (fitna) and you both are able to openly practice the rituals of your religion, then there is no objection to your travel with your wife for her to pursue her studies. However, if the studies involve unlawful matters, or if residing there presents a temptation (fitna) to one's religion, then neither traveling nor studying is permissible.