To show fog movement in a single photo, you generally need some motion blur, so a longer exposure is the main technical tool. If your results look either too static or too blurry, the issue is likely that you haven’t found the right shutter speed yet.

Composition matters too: fog is easiest to read when it interacts with something. Look for edges where it wraps around a tree, building, person, or other subject, because that shape helps suggest motion. Without a reference object, fog can just look like a soft, featureless blur.

You can also emphasize movement by capturing distinct shapes or flow patterns in the fog rather than trying to record all of it evenly.

That said, a single still image has limits. If your goal is to show the evolving motion itself, a time-lapse may be a better fit than one frame.

In short: use a longer exposure, experiment to find the shutter speed that gives some blur without losing all structure, and compose the shot so the fog’s edges and interaction with objects are visible.