Let me say this first: A cold is a serious condition for a turtle.
Turtles get colds, just like people and most other animals. They may sneeze, but sneezing is not always a sign of a cold; dust or just "stuff in the nose" can make a turtle sneeze occasionally. The turtle may have a bit of a runny nose, too. Again, this can be causes by allergies, or by a cold. If you are not sure, whether your turtle has a cold, have a veterinarian evaluate the animal.
These symptoms could also be a harmless little cold, or they could be the only sign of the beginning of a more serious respiratory infection, which must be treated with medication.
While a turtle doesn't get a cold from being cold, keeping a turtle too cold, or under less than ideal conditions, will weaken its immune system, and it is more likely to catch the disease.
Colds, respiratory infections, and pneumonia can be caused by bacteria or viruses. Just like in people, if a bacterium is the agent, antibiotics are used for treatment, and there is a good chance the animal will recover. If a virus is the cause, which is hard to determine, since it won't show in a culture, nothing much can be done, except to keep the turtle warm and wait and hope.
The first measure for any cold-like symptoms is to raise the temperature in the turtle enclosure. Raising the temperature helps the turtle's immune system become more active. Also, make sure the turtle has the cleanest possible environment. How much to raise the temperature? Maybe a few degrees for water, to around 82-85 degrees. Better even, is to take the turtle, if it is a water turtle, out of the water. Put the turtle into a heated box--use a thermometer! make sure it doesn't get too hot!!!--you can use a heating pad or a lamp to heat the box. Leave the turtle in the box except for two soakings of 1/2 hour each mornings and evenings, for eating and drinking.
If the turtle gets worse, or if it does not improve within a couple of days, see a veterinarian immediately.
If any of the symptoms are more than minor, or if the turtle also gasps, has a rasping breath, or swollen eyes, is sluggish or doesn't eat, see a veterinarian immediately.
If you are not sure--see a veterinarian. Colds, respiratory infections, and pneumonia are probably one of the most common causes of death for turtles.
Treatments for colds, respiratory infections, and pneumonia include warmth, antibiotics (oral or injected), steaming (with or without medication), breathing oxygen (if the turtle has water in its lungs and cannot get enough oxygen), and X-rays (to check for water in the lungs).