So I am required to pay taxes for the time I did not live in my House?
Unfortunately, yes because the taxes stay with the home. You can contest it, if it was never disclosed by the agents. You may be able to recover. Talk to a Real Estate Attorney.
The taxes have to be paid and if the prior owner did not then you are required to. If it was not in your contract for the previous owners to pay the taxes through the closing date you are totally on the hook. If your contract states they have to pay you better pay and try to recoup so Fairfax County doesn't foreclose.