Yes cancelling your auto insurance or other insurance policy does affect your no claims discount and with most insurance companies you'll lose your no claims discount if you do cancel your insurance policy.
You still may be able to get a refund for the rest of the remaining months left on your insurance policy when you do cancel the insurance policy but it can depend on what is laid out in your terms and conditions.
It's best to not cancel your insurance policy whether auto insurance, homeowners insurance, renters insurance etc unless you absolutely need too and can find a better insurance company with cheaper rates.
Although just because an insurance company has cheaper rates doesn't mean they are a good insurance company so sometimes it's best to stick with your current insurance company.