Can moving cycling cleats forward be a good way to train our calf muscles?
There's no point.  Calf muscles do not provide significant power in cycling:

Further down the chain into the lower leg the gastrocnemius and soleus haven't been found to add much power to the pedal stroke but their main role is to stabilise the lower leg to enable an efficient transition of the force generated by the upper leg to the pedal.

The final part of a stride while running is the calf muscle extending the foot to get a final push off the ground - and this happens when the leg is just about straight, so the force that gets transmitted pushes directly on the center of mass and helps propel the body forward.
The power stroke when pedaling occurs with the knee bent.  Any force the calf muscle exerts just gets transmitted through the knee to the quadriceps, which still have to do the work to push the knee down and provide power to make the bike go.
Lengthening the lever arm the calf muscles have to work against by moving the cleat forward on the foot just makes the calf's muscles work in stabilizing the ankle and foot while pedaling harder without generating any more power.