Can a toothache be related to being sick? 

Short answer to this is: no. Every source I have found came to the same conclusion. Toothaches are only related to teeth; in other words, they do not reflect pain or injury in another body part. 
If you do have toothaches, there are a few things you can do to alleviate the pain. According to top10homeremedies.com, Dental problems such as toothache and bleeding gums can be easily cured at home with the use of asafetida. Simple add a pinch of powdered asafetida to lemon juice and then slightly warm it. Now with the help of a cotton pad apply this solution on the affected area. This will give you quick relief from the pain. Alternatively, asafetida fried in clarified butter can also be applied in the tooth cavity for instant relief. 
Another remedy mentioned on the site is salt and warm water. One effective way to treat toothache is to use warm salt water. Mix half a spoon of common salt to a glass of very warm water and use this solution as a mouthwash to rinse your mouth thoroughly. When this solution touches the infected tooth it will extract fluid from the tissue and reduce the swelling on the nerves. This mouthwash will also take care of the bacteria that cause infection. 
Furthermore, there are some remedies that are more commonly known, like cloves for toothaches. No home remedy for toothache can be better than clove. Clove has got anti-inflammatory, anti-bacterial antioxidant and anesthetic properties that help in alleviating toothache to a great extent. At the first sign of toothache, dab some clove oil with the help of a cotton ball directly on the infected tooth to get some relief. You can even mix a few drops of clove oil in half a glass of water and use this solution to rinse your mouth. Else you can even grind some whole cloves and mix them with a few drops of cooking oil before applying it on the infected tooth.Cloves are amazing. I can say that from personal experience. They numbed the pain so much that I was able to eat normally right away. 
Garlic is another option: Use of garlic can also give immense relief from toothache. Garlic has got antibiotic properties and other medicinal properties that can be very effective in slowing down bacterial affects. Simply crush a clove of garlic and add some rock salt and then apply it directly on the affected tooth to alleviate the pain. Instead of fresh garlic you can even use garlic powder. If possible, you can even chew one or two cloves of garlic to get relief from toothache. Repeat this natural treatment for a few days.
There are lots of other options like onion, wheatgrass, bay berry water, water and peppers. But no matter what you do, remember that going to the dentist is ultimately going to be the best way to get of the pain. Doctors know best, after all, right? 
