No, you do not owe a deductible amount if the person that you hit files a Property Damage Liability claim with your car insurance company for the damages you caused them.
Neither you nor the person you hit has to pay a deductible for the damages you caused when they make a claim with our Liability coverage. Deductibles are due when you use coverages for which you have chosen a deductible amount for. This normally includes both physical damage coverages of Collision and Comprehensive. In some states, a deductible is applied when you file against your own Uninsured or Underinsured Property Damage (only available in a few states).
If you have collision coverage on your car and have to place a claim for the damages your vehicle sustained in the accident that you were at-fault for than you would be responsible for the deductible amount associated with your Collision coverage.