No, you cannot negotiate car insurance rates because the industry and prices are heavily regulated by each state. An insurance company can't change its range of rates, once it is approved, without a new state review and evidence that the change is financially necessary. If your insurance company gave you a significantly different rate from the first one it quoted you, the insurer could face large fines or lose its insurance license.
Although you can't negotiate insurance rates, you can strategically negotiate the insurance shopping experience to get the lowest price possible for the coverage you need. While every insurer's rates are approved by the state, they aren't all the same. Comparing car insurance quotes from multiple companies is one of the most important strategies for finding the best deal. Taking full advantage of discounts is another. Discounts don't change the base rates; they offer a percentage off the total premium. Companies often have many discounts that you may not know about. For example, GEICO has affiliation discounts for belonging to, or working for, more than 800 organizations. It is worth a call to a customer service representative to see if there's a discount or two that could help lower your premium.
You also have the options of lowering your coverage or raising your deductibles to get your premiums down. You must have the minimum coverage mandated by your state, but anything above that is your decision. Think carefully about what you need. If you have an older car, for example, you may not even need collision/comprehensive coverage. Also consider your deductibles carefully. Raising them can reduce your premium, but you don't want to make them so high that you can't afford to pay them in case of a claim.