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?

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.)

Fog Isn't An Object

As has been stated elsewhere, gasses and liquids are not considered objects. So fog would not be a valid image created by Minor Illusion.

There could be some edge cases where physical inspection might not reveal an illusion (for example, an illusion of a brick in a wall painted green, cast a nanometer in front of a real brick in a wall painted red, that would feel the same to a person investigating it), and a DM would have to decide whether to ignore that part of the spell's description. However, creating illusions of intangible things would normally be impossible with minor illusion.

_{NOTE: If the spell in question had been silent image (which contains similar text about physically interacting with the illusion, but allows the spell to create most any "visual phenomenon"), then see Ryan Thompson's answer on how physical interaction could still reveal illusory fog.}

In my experience of fog, you can see it a ways away but as you near fog, you realize that there is no point in time that the fog fully obfuscates your ability to see. So if you were to walk up to and into illusory fog, it would continue to block your sight. REAL fog doesn't.

So being aware that real fog doesn't block sight as well as fake fog would, you would realize that it is an illusion, and thus, see through it.

Pun intended.