I believe there is a sheet that itemizes $ of a property purchasing. I remembered there was such a sheet during my recent refinancing. Basically it was like I sold my property to myself. There were some adjustments to payment that my new bank is responsible for, and some of them are tax and escrow. I think, the new bank reimbursed the old bank for some part of the already paid property tax in my case.
If the seller still has mortgage, it's unlikely you get into this situation, because the seller's bank holds an escrow account for the seller and it has always enough fund to cover property tax. Read the closing financial documents and see how the tax was handled. There is a chance that seller's bank gave the tax to you rather than the county at the closing, so you could pay the tax for the next statement.
If you have evidence that the seller didn't actually pay the way like the closing document stated, you certainly can ask them to pay you back in my opinion.
