Are there any statutes of limitations on insurance claims?

Statute of limitations varies by state, most are 2 years, according to this article on The Zebra CA gives you 2 years.

Whether or not the statute of limitations has passed, lying about the specifics of the claim would be considered insurance fraud, not worth committing fraud over a car.

Your contract probably has a maximum period of time for you to report an incident and file a claim. This is typically a short window that can be regulated by your local state government. I've heard it as low as seven days. It's unlikely that your state would force a deadline as long as a year.

Your last couple of sentences are about insurance fraud; I'd avoid insurance fraud.

"I could very easily say the damage happened 3 months ago and I doubt they would be any wiser about it."

The last time I had my car in for repairs under an insurance claim they made sure that nothing I was claiming was showing rust. They wanted to make sure I wasn't tacking on other older damages to the claim.

Don't give incorrect information regarding the damage, the date, and the cause. You don't want them to cancel your policy or charge you with insurance fraud.

The state government controls insurance law. I checked with my state insurance website and couldn't find anything that talked about a statute of limitations. I was able to see information regarding a limit on filing a lawsuit. I have no idea if they are the same.

Something else to consider is that making the claim may cause your policy to increase. Make sure that you understand your deductible level on your policy.