"Know of any" is correct; you probably heard it before and picked it up.
"Know of" is better than "know" here, because to "know of" is to be aware of the existence of something, which is what you are asking: do there exist any coaching centers in the area.
? Do you know of any good dentists in the area?
This is somewhat questionable. You cannot know whether a dentist is good or not simply by being aware of their existence. So this is essentially asking: have you heard from other people that there are good dentists in the area, and do you remember who those dentists are?
If someone tells you that he has been to a good dentist and gives you the contact info, you thereby know of a good dentist. But you do not know that dentist.
Do you know of any dentists in this area?
= Do you know whether there exist dentists in this area and where? (Have you seen a dentist's business signage on a building somewhere around here?)
Do you know any dentist in this area?
= Are you familiar with a dentist in this area? (Is there a dentist here you have visited?)