Is the distance between Qalyubia and Cairo considered a journey that necessitates the presence of a mahram for a woman whose father has died and whose only brother is a student who only comes home at the weekend?

A woman is not permitted to travel without a mahram (definition) (unmarriageable kin). This includes both long and short journeys according to the majority of scholars. Everything that is customarily called travel is forbidden to a woman unless she is accompanied by a mahram. Al-Nawawi held that travel is not restricted to a specific distance; rather, it encompasses everything that is called travel.

Leaving Qalyubia for Cairo is not considered travel by custom. Therefore, there is no harm in going there to fulfill needs without a mahram.