The answer in most cases is yes, the new auto insurance company will find out about your claims history. The new auto insurance company will want to know who your last auto insurance company was. If you tell them you have not had insurance for a while, that will have a negative result on your new auto insurance rates, so you really have not gained much by trying to switch auto insurance companies. Also, most reputable auto insurance companies are locked into a database that tracks auto insurance claims on a national basis. And lying to another company will merely result in a denial of a new policy or even higher rates.
That being said, it still doesn’t hurt to shop around for cheaper auto insurance rates. Be sure to review proposed policies, however, to make sure you are the coverage you need from other auto insurance companies.