You are generally not required to give a recorded statement to the other party's insurance company. Your own insurance policy is different: most policies contain a cooperation clause that obligates you to assist your own insurer, so the answer depends on whose adjuster is asking.
The other driver's insurer
That adjuster represents the company paying the claim, not you. A recorded statement taken in the days after a crash is taken before the full extent of an injury is known, and an early description such as saying you feel fine can be used later to argue the injury was minor or unrelated.
Your own insurer
Most policies require cooperation with your own company, including for uninsured and underinsured motorist claims. Declining outright can create a coverage problem, so this is the situation where reviewing the policy language first is worth the delay.
Why timing matters
Some injuries, particularly soft tissue and head injuries, are not fully apparent immediately. Describing your condition before it has been evaluated commits you to an account you may not be able to correct. There is normally no obligation to answer the same day the request is made.