As @FumbleFingers said in comments, the most likely use the preposition with:
We need to make an appointment with a dentist.
Also, we often use the direct article to mean either "my regular dentist" or even "any dentist in general":
We need to make an appointment with the dentist.
We'd most likely use the indefinite article a in a case where the person making the appointment does not regularly go to the dentist.
But really, the most natural usage is not to use a preposition at all. We'd use either a possessive or a noun adjunct:
We need to make a dentist's appointment.
We need to make a dentist appointment.
Either form is correct - I would personally use the possessive, but the non-possessive looks natural to me as well (and in rapidly-spoken English, you can barely detect the difference between the two).
Interestingly enough, the popularity of the possessive is different for different medical professionals. For dentist, using Google Ngram, it looks like the non-possessive has been the most common usage since about 1965:
But for doctor, the possessive is by far the most common (in fact, "doctor appointment" sounds unnatural to me):
I am guessing that this is purely a matter of pronunciation: it's difficult to pronounce the [sts] cluster at the end of dentist's distinctly, so people just leave off the final [s] and that has migrated to written English as well. The [rs] at the end of doctor's is easy to pronounce, so it has remained the preferred usage.