If your symptoms have persisted since your accident three months ago, I would recommend seeing your primary care physician and possibly being referred to a neurological surgeon. You may require MRI imaging and plain film X-rays to further evaluate your spinal condition.
Jason E. Garber, MD, FAANS, FACS
An isolated back injury such as a fracture or herniated disk can cause a neurological deficit but it would affect the legs only, not the hands.
The best way to proceed here would be first to see your doctor to obtain a full history and neurologic exam. If neurologic deficits are found (weakness, sensory loss, etc.), then imaging studies (CT/MR scans) might be appropriate.
But the simple answer to your question is a lumbar spine injury would not cause problems with the hands.