Yes, no , maybe

Is it totally the other drivers fault. Did the other driver have insurance and will that insurance pay your claim completely ?

If you are not at fault and your company does not have to pay , no it should not go up.

If the other driver does not have insurance and your company pays for your damage, often yes it may

But it will depend on a company by company issue, some may, some will, and some won't for the first accident

It can if the other driver doesn't have insurance and you have to file a claim. Some auto insurance companies will offer you a grace and not raise your rates after the first accident depending on your circumstances.
http://www.smartautoinsurance.com/ has a lot of information on auto insurance. I reference them a lot especially for quotes. Good luck!