Minor illusion says:
If you create an image of an object – such as a chair, muddy footprints, or a small chest – it must be no larger than a 5-foot cube. The image can't create sound, light, smell, or any other sensory effect. Physical interaction with the object reveals it to be an illusion, because things can pass through it.
You could create an illusory fog cloud in a 5-foot cube. However, things pass through real fog clouds just as easily as through an illusion. So how exactly does "things passing through it" reveal an illusory fog cloud to be illusory?
Best Answer
Real fog still interacts visibly with things passing through it
If you pass your hand through a real cloud of fog, the fog will visibly flow around your hand, and the air current produced by your hand will cause the fog to swirl around after your hand passes through it. An inanimate illusory cloud of fog will not exhibit any of these effects, revealing it as fake. In addition, depending on what the fog is made of, it may have other associated sensory effects, such as dampness or coolness, none of which can be replicated by minor illusion.
(Also, as another answer points out, minor illusion in particular cannot produce the image of a cloud of fog.)