Unless you're filing a claim against a government agency or employee, you need not notify the people you think are responsible for your injuries within a set number of days after an accident. But that doesn't mean you should drag your feet. On the contrary, acting right away -- within a few days, if possible -- will increase your chances of receiving a quick and fair resolution to your claim.
Giving notice doesn't obligate you to file a claim; it simply preserves your rights and prevents others from later saying that your claim is unfair because you waited too long to tell them about your injuries. If you promptly notify others that you intend to file a claim for your injuries, you can then move at your own pace in processing and negotiating the claim with the insurance company or government agency that winds up taking responsibility.
For more information, including whom to notify and how to give notice, see the article Personal Injury Claims: Notifying Responsible Parties.