Is she required to make up the fast (Qada') or offer expiation (Kaffarah) if she broke her fast in Ramadan due to traveling a distance of one hour, believing that breaking the fast was permissible based on the Qur'anic verse, "But whoever among you is ill or on a journey, then [he must fast] an equal number of other days," knowing that she is 18 years old, and is it necessary to make up the fast before the next Ramadan?

It is not permissible to break the fast without consulting scholars and verifying the ruling of the Sharia. The majority of scholars do not permit breaking the fast except for a journey that spans four burd (approximately 83 kilometers).

If the distance you traveled was less than that, then you must seek forgiveness for breaking the fast without verification. You are not considered to have intentionally broken the fast due to misinterpretation, and you must make up this day. No expiation is required of you, except for breaking the fast through intercourse, according to the more authoritative opinion.