Yes, actually, they do.
To be more precise, grey horses are at higher risk than other colors for melanoma. This is because the greying mechanism locks melanin out of the hairs and concentrates it in the skin. If you look at a grey horse close up and part the hairs, you'll see that its skin is black or very close to. Most horses have medium to dark grey skin. (Pink skin is seen on highly dilute horses and under white markings). (A few grey horses also have skin that lightens with age).
However, "Grey horse melanoma" is generally not serious. Yes, it's cancer, but it grows slowly and seldom spreads beyond the skin. It is most often found under and around the base of the tail, but is also seen around the genitalia, lips and eyelids. In some cases, these lumps may be surgically removed (especially on harness horses where they can interfere with placement of the crupper). However, the vast majority don't spread and don't even seem to bother the animal.
Rarely, however, grey horse melanoma can become malignant. They may spread into the lymph nodes and lungs. It can be hard to predict, and rate of growth is generally the only way to tell the difference.
The cancers generally show up in horses that are older, at least 15 years old. If grey horses happen to be prized in your culture, then "age lumps" might be a thing that would be discussed. Melanoma does also show up in non-grey horses, but is much less common.