Tested it for certain and no, using Calm Mind does NOT have any effect on Focus Punch.
In this replay here, Tyranitar 1 hit No CM Medicham during Focus Punch, like usual. Even after CM Medicham used Calm Mind, it still had to Focus before using Focus Punch on Tyranitar 2. It was a nice thought, but it just doesn't work like that in the games, sorry.
Also, even if it did work, I wouldn't really recommend doing this strategy. Calm Mind boosts the user's Special Attack and Special Defense, while Focus Punch is a Physical move. Wasting multiple move-slot in order to use 1 Attack while boosting a different Attacking stat isn't really worth it. It's better off to use a different Fighting move, so you don't have to worry about wasting turns to pull off 1 Attack. Besides, if you get hit during Calm Mind, don't expect your opponent to forfeit, wondering why you're setting up so much instead of attacking.
Hope this made sense and helped you :)