Can I receive a property tax deduction or credit if I moved in the middle of last year??

If you are referring to the property tax on a property you bought or sold during the year that you did not pay the tax bill for, the answer is yes.  If you sell a house before the property tax bill for that year is paid, there will be an adjustment on the closing statement where you are assessed your share of the property tax.  So if you sold a house on July 1, and the property tax had not been paid at the time of closing you pay the buyer for one half of the property tax that he was going to have to pay later in the year.   If you had already paid the property tax then the buy would pay you for half of the tax.  It will be on the settlement statement.