Slow and Steady...
I hear you saying, "Play a Board Game with a Turtle Day? Huh?"
I'm with you there. This one is "out there" quite a long way.
Maybe it's like Play a Board Game with [Someone Dressed Like] a Turtle Day. If so, get out your old TMNT costume, have someone put it on, find a game, and play.
There's a small possibility that this day refers to the old educational computer language called Logo that used "turtle graphics" for drawing on the screen. But even that reference doesn't involve playing a game.
It could be that you're supposed to find a board game that incorporates a turtle as one of the bits and play that game with someone. I can't think of such a game though. Can you?
On the slim chance that you actually do have a live turtle - maybe a pet - in the house, you could simply attempt to play a game with him. You'd have to take his turns for him, make his moves for him, so it would pretty much be playing solo.
If you played both roles equally well, the turtle might even beat you.
Hare and Tortoise (which I used to own) would be an excellent game to play, though even in this one you play as the Hare, not the Tortoise. Over 4000 people do own the game. Are you one of them?