Is trigeminal neuralgia the same as trigeminal nerve injury caused during an accident ?

No. Injury means acute damage. Neuralgia (nerve pain) can be from such things that irritate the nerve or diabetes or arteritis.

An injury to the trigeminal nerve would tend to result in numbness and/ or weakness over the specifically involved branch, and very unlikely to cause nerve pain. Trigeminal neuralgia can result in intense attacks of sudden lancinating immobilizing pains due to mild nerve compression at the brain stem level. Multiple Sclerosis can cause, but even compressing blood vessels may provoke.