Certain volatile contaminants in water are more noticeable (as regards their odor) in hot water, because when the water is heated these contaminants vaporize and are released into the atmosphere.
Oxides of sulfur and other chemicals behave this way. Your well is probably drawing sulfur-contaminated water and you are just noticing it in your hot water for the reason stated above. The sulfur is certainly also in your cold water, you just haven’t appreciated it yet.
Have the water analyzed in a lab. You have three choices : 1) live with it (sulfur is stinky but not toxic);
2) Install an water purification system; or, 3) drill a new well (not guaranteed to solve the problem…you may still get sulfur in your water).