Would my testimony be considered false if I testify on behalf of a colleague that she fell at school, even though I don't precisely remember the location of her fall, and would expiation be incumbent upon me if her claim is proven to be false?

A Muslim is not permitted to testify except to what he truly knows to have occurred, based on the Almighty’s saying: "And we did not testify except to what we knew." Therefore, if you helped your colleague carry her belongings and learned from her and from circumstantial evidence that what happened to her occurred at school, then there is no harm in testifying for her based on what you came to know through evidence and circumstantial indicators. However, if you know nothing about her and merely sympathized with her, then you are not permitted to testify so that she may benefit from this testimony, as this is forbidden false testimony, which is one of the grave sins. If you have testified to what you do not know, then you must hasten to repent and resolve not to repeat such an act.