Can an insurance agency find out about a claim or accident even if there was no police involved or report on it when asked about your past accident history?

Yes, CLUE reports monitor anything paid by any insurance company regardless of a police report. Therefore, only if you reported the accident to the insurance company or they reported it. If you did not, and they did not, and there was no police report(for example you did some kind of direct negotiation or something) only then is there no way they are going to find out.