If you work for the city or state, then you may still be included in the standard workers' compensation system. But there may also be a special statutory rule that applies only to public employees. Basically, if you are a public employee then special rules may apply to you, so make sure you do your research before filing a claim. You don't want your claim denied because you missed a few rules or filed with the wrong agency.
If you are a federal employee, there are several different programs that may cover your injuries. The Federal Employee's Compensation Act allows you to recover medical costs and lost wages for any injury that disables or kills you that was suffered while on the job. These protections cover all federal employees from federal officers to meat inspectors. These claims are managed through the Office of Workers' Compensation Programs. It is probably a good idea to speak to an attorney first, but if you are a federal worker, then you will likely file a claim with this office.
If you are a state or city employee, then there are probably local rules that govern your workers' compensation claims. You will probably report any injury or claim to your local human resources office, which at the very least can direct you to the proper agency. Most cities and states require that employers hang posters that explain how a workers' compensation claim works.
If you were injured while on the job, then you may want to speak to an attorney. If you also happen to be a government employee, be sure to mention that to your attorney. It is unlikely to affect whether or not you are entitled to a claim, but it will likely affect how you get your claim filed.