The Fog Cloud cannot be colored, RAW
Without circumstances dictating otherwise, spells in 5th edition D&D only do what they say they do, and no more. The Fog Cloud spell says it creates a 20' radius sphere of Fog, and that's all you can guarantee it'll do. If you want it to do more than that, you'll need a different spell.
You should be able to combine Fog Cloud, Light, and (optionally) Mage Hand to create the appropriate effect
Placing a colored light source inside the Fog Cloud should diffuse the light through the fog, creating the colored Fog Cloud of your choosing. Since of those three spells, only Fog Cloud requires concentration, it's possible for the same spellcaster to maintain all three effects simultaneously, and both Light and Mage Hand are cantrips, meaning you're not spending extra resources (other than your Action Economy) to produce this effect.
Simply cast Light (for the color of your choosing) on a rock, then use Mage Hand to lift the rock into the fog cloud (or simply place the rock inside the fog cloud if your character is physically able to do so). The range on Light extends at least 20 feet, and while it wouldn't disable the Heavily Obscured effect, it is plausible that a character could generate a colored fog cloud using this method.
Bear in mind that you may need a permissive DM to make this work, RAW, since the fog will surround the light, and while this should not obscure the light source itself, it's not clear that the physics of the light diffusing through the fog will result in the fog itself taking on the light's color. So it does require your DM to allow for real world physics to take precedence in this situation.