Your car insurance would not cover the damage to the garage, only the damage to your vehicle if you have collision coverage.
As for your second coverage, I would recommend paying for the damage out of pocket. Making a claim on your homeowner's insurance will increase your rates, most likely more than what it cost to make the repairs. Keep in mind your deductible will apply. Most deductibles for homeowner's insurance are $500-$1500. So the insurance company would just file the claim and you would still be responsible for the damage because it does not exceed your deductible. Long story short, avoid the hassle and pay for the damage yourself. Good luck and if you have any other questions, don't hesitate to ask.