Can using cold water relieve toothache?

No: It very well may make pain worse, as sensitivity to temperature is a hallmark sign of tooth pathology.

Maybe: Some dental infections create pressure inside a tooth. Cold water can relieve pain and heat can increase pain. If you notice this to be the case for you cold water can be used until you can see a dentist for permanent relief. Ususally you will need root canal treatment or an extraction if you have an infected tooth.

Tooth pain: If cold water is relieving your toothache, you need to go see a dentist. This either means you have a lot of swelling within the mouth and gums, or your nerve is irreversibly inflamed. Go see you dentist for evaluation and treatment before you get more swelling and or infection. Chances are your tooth can be saved if you don't wait too long.

Temporary, perhaps: The ONLY way to cure a toothache is to remove the infected tooth material (cavity treatment, root canal, or extraction). Some infected teeth respond temporarily to cold, but it's not a cure. Please don't delay, get professional help now before the infection spreads any farther.