Not for the larger surveillance drones, no. Those are a km up, you are unlikely to even know it’s there.
For small drones a shotgun is great… if it’s semi or full auto, if the soldier knows exactly where a drone coming right at him is, and has time to prepare while it’s still around 50 yards out.
Outside of those conditions, still not great.
Shotguns are roughly “effective” up to 100yards with slugs, and half of that with buckshot. Drones dropping grenades are generally well above 200 yards up, generally more like 500. So it’s really only the suicide ones that can be reasonable targets. The problem is a purpose build FPV drone typically used in suicide runs can haul ass, and cover 50yards in a second, are hard to see and predict the path of, even when you know one is there. Consumer drones can generally cover 50 yards in roughly 2-4 seconds.
Trying to shoot one moving across your view dive bombing on something else is going to have you more likely accidentally shooting the drones target or something unintended rather than the drone itself.
I do anticipate vehicle mounted scattergun systems meant for close in defense of drones though, as well as quickly/auto launched anti-drone drones for further off defense. More powerful drones would be needed for dealing with stuff 1km up and beyond though, the smaller stuff will struggle getting close to that.