If you getting a mortgage expect to pay a full years taxes in advance at closing. Your attorney can give you a better idea on what you will be paying it may be just a quarter. You can also call the mortgage company and ask how many months of taxes they require in escro.
Next to your question you will be responsible for the number of days left in the quarter. This should all be detailed on the HUD 1 which should be provided to you just before closing.
Like I said this is best answered by your attorney. Also if your working with a real estate agent they should be able to help you get the answers to these questions for your exact situation