I don't know whether your insurance company will raise your rates, cancel your policy, or what. The problem is that unlike the old days a report is required by law to the DMV. That information is going to be available to your insurance whether you turn the claim into them or not. Can you get away without filing the report? I couldn't tell you. If you don't and your neighbor does it might become an issue for you.
This is from the California DMV
California law requires traffic accidents on a California street/highway or private property to be reported to the Department of Motor Vehicles (DMV) within 10 days if there was an injury, death or property damage in excess of $750. Untimely reporting could result in DMV suspending a driver license. Accidents involving vehicles not required to be registered such as an off-road vehicle (OHV), implement of husbandry, or snowmobile or occurring on a military base or occurring on the driver’s own property involving only the personal property of the driver and there was no injury or death are not reportable.