No. Buttermilk doesn't automatically make a dry recipe moist. Substituting it for whole milk in equal volume means you actually have less fat in the recipe since buttermilk is typically low fat. Fat is a tenderizer. For many recipes that use baking soda, buttermilk creates an acidic medium so that baking soda can be neutralized--if milk was used instead you'd have an awful tasting cake from all that unneutralized baking soda.
The advantage of substituting buttermilk for milk (all necessary acid-base adjustments applied) is that its acidity inhibits gluten formation so it has a tenderizing effect. It also has that characteristic tang that many like. Subbing it for the milk in the recipe you have up will most likely give you a different flavor profile more than anything else. It can't hurt to try.
Dryness in a lot of cakes is often due to inaccurate measurement of the dry ingredients. Do you weigh your flour and cocoa? If not, you should get a kitchen scale and measure the weight of ingredients called for. This ensures you're not using more flour or cocoa than called for which would dry out your cake. It also ensures consistency between each batch you make.
Also be sure not to overbake. Even the best recipe is going to dry out if left in the oven too long.
As a last resort, if you're convinced your cake is fine and the client just prefers a 'wetter' cake, you could brush the layers with simple syrup when you're assembling each tier.