For how to request the time off, as others have already said, just say it's a doctor's appointment without specifying whose or its nature, or say you need some personal time. Generally, such requests are respected without further inquiry, at least in my experience.
Something not said so far is that you could offer to make up the time if the nature of your job and your workplace (and your schedule) allows. Thus you might be able to avoid using vacation or sick leave (or the more generic paid time off if that's what your employer grants). You might want to tell the boss something like "I need to be out of the office on Thursday afternoon. If it's okay with you, I will make it up by coming back into work that evening." Or whatever works for you. You may even find that - if your job allows and you're not otherwise missing meetings or other important work events - you don't need to bother saying anything, although I don't recommend assuming that, especially when you're new. In my job I can often just leave for whatever non-work event I need to be gone for and make up the time as my schedule permits.