This "Gem Cold" is an illness that the Crystal Gems had contracted in the Season 4 episode "The Healing Crystal". There isn't too much known about the illness, including the exact name for it, as a name was never given.
Cause[]
It is unclear of what exactly caused the illness, but it's implied that it can be contacted by one of Quetzalcoatl's feathers when they begin to decay and turn to dust.
Effects[]
Apparently the effects of this is similar to a common cold: typically giving the victim a runny nose, and prone to sneezing fits. The gemstone on a Gem starts to become liquefy, becoming similar to a water balloon, and making the Gem a lot weaker. It's unclear how dangerous the illness is, nor if it's restricted to only Gems.
Cure[]
The only known cure to this illness is with the Healing Crystal, a type of artifact known to cure the illness by placing it on a Gem's Gemstone. It's never explained how it worked or what the exact cure is from it.