As far as I’ve seen, there’s never more than one birthday on any given day, and same with an event and a birthday on the same date. So like for an example, Saint Cethleann Day and a birthday haven’t coincided. However, when a birthday falls on a free day or a mission day, which I’ve seen happen before, you get the prompt to give flowers/host a tea party before being taken to the mission select/free day screen.