If I see a grouse while I'm deer hunting, do I have to have a shotgun to shoot it?

No. The upland game rules offer an exception for forest grouse to address this exact situation. Unlike other birds, you can take a forest grouse with a center-fire weapon, a sling shot, or even a stick or a rock. It is still not legal to take a forest grouse with a trap, snare, net or crossbow.