Does pouring water on burning charcoal when grilling make the meat more smokey?

No, as that isn’t creating smoke. It’s creating steam. A steamier cooking environment may actually inhibit desired grilling flavors as it could inhibit browning via the Maillard reaction.

Do not do this. You will rapidly cool and may even extinguish your fire. This will also cause ash and cooler, more astringent, elements of smoke to land on your food. The results will be very unpleasant.

If you want your meat more smokey, you just need to apply the appropriate techniques for smoking. Grilling imparts some smoke flavor, but that is generally not the primary intended outcome.

The "smokey" taste does not actually come from the charcoal - it is produced when the juices from your food drip onto the charcoal, turning into highly fragrant compounds that then waft up with the smoke and stick to your food.

So no, just adding water to the charcoal wouldn't help reproduce the "smokey" taste.