To "suffer an injury" means to get injured (or, simply, to get hurt). The phrase "suffer an injury" is formal, and mostly used in news, in legal documents, in history textbooks, and so on:
Employees who suffer an injury as part of their working duties are entitled to compensation.
In addition to the word "injury", you can also suffer:
- an illness
- a fracture (meaning a broken bone)
- a heart attack
To "suffer" an injury is different from "suffering from ___". The first phrase just means to get hurt. It doesn't mean that the injury was necessarily painful. The second phrase means to feel pain and discomfort over a period of time. It's used for talking about longer-term illnesses and conditions:
Both my mother and my grandmother suffered from diabetes.
Do you suffer from allergies?