Property taxes are paid twice a year. The first installment is due November 1 and are delinquent December 10th. The 2nd installment is due February 1st and are delinquent on April 10th. If you closed in December your taxes will be pro-rated from January to February and will be paid through escrow. The seller will recieve a credit for the days that you pay from the close of escrow date. Escrow will take care of properly applying the credit and payments to the buyer and seller. You cannot close escrow without these payments/credits being made.
Be aware that you will most likely receive a supplemental tax bill after escrow closes. The county assesor will re-assess your property based on the new property value vs the old property value. This bill will also be pro-rated based on your close of escrow date and the next tax installment due.