Welcome to the forum!
. . . Now in his defense, we currently have a mole problem and I think that's what is getting him to dig. But how would I communicate that it would be ok to dig in our natural areas or backyard even, just not in the front. . .
You are probably right, in that the mole problem likely prompted the situation. The scent and/or sight of moles has to be hard for a puppy/dog to resist.
Two tips that seem to be shared a lot here are:
Don't give the puppy free access to the area you don't want them to dig in. Either have the puppy on a leash (with someone in attendance); or if possible, tether the puppy in such a way to restrict access while in the front yard.
Make a sand box area in the back yard, and encourage the puppy to dig there (perhaps by hiding some treats in the designated dig area). You could make a raised planter box or use a kiddie pool that you fill with sand.
I'm sure your trainer will likely have some suggestions. Do you have a plan to get rid of the moles? I'm thinking that as long as the moles are there, you will have to keep your dog out of the front yard. You could get the dog to dig in a designated area in the back yard, but I'm thinking that if the dog has unsupervised access to the front yard, the digging will continue.
My puppy is 7 months now. She did do a lot of digging a couple of months ago. She was preoccupied with burying chew treats, and then appeared to sniff out areas that a cat had buried poo and did some digging. She would even put her paws in her water dish and dig out the water, LOL
We are both still a "work in progress". Since I quit giving her bones, she had done less digging and burying. She does want to hide her dental treats, but there is a lot less digging then there was with bones.
I still have to take precautions with her water bowls. To prevent unexpected floods, her current main water source in the house is a bucket of water, kept inside the bath tub. I went from large stainless steel bowls to placing the bowls inside a weighted down 8" plant pot, then a bucket on the bathroom floor and now a bucket inside the bath tub. At the moment, it's unlikely that she can lift the bucket (filled with water) out of the bathtub and dump all the water on the bathroom floor again (and flood the hall as well). It will be a balancing act to keep enough water in the bucket, so she can't lift it out of the tub, but not to much, should she jump in the bath tub and start digging water.
In addition to replies to this thread, you might want to view some older threads. While viewing your thread, if you scroll down towards the bottom of the page, you will see some links to older threads on "digging" that could be helpful.