Belly Drum Azumarill, in my opinion, is the second scariest mon after power herb geomancy Xerneas, and thinking of something wasn't easy.
Water/Ground Counter
Quagsire @ Leftovers
Ability: Unaware
EVs: 252 Def, 132 SpA, 124 Spe
Bold Nature
- Recover
- Sludge Wave
- Scald
- Toxic
Quagsire's ability to deal with Belly Drum Azumarill lies in unaware. This makes the Azu do damage to it as if it had never belly drummed. Max Defense EVs allow it to take the hits well:
252+ Atk Azumarill Knock Off (97.5 BP) vs. 4 HP / 252+ Def Unaware Quagsire: 52-62 (15.6 - 18.6%) -- possible 8HKO after Leftovers recovery
252+ Atk Azumarill Aqua Jet vs. 4 HP / 252+ Def Unaware Quagsire: 33-39 (9.9 - 11.7%) -- possibly the worst move ever
252+ Atk Azumarill Play Rough vs. 4 HP / 252+ Def Unaware Quagsire: 72-85 (21.6 - 25.6%) -- possible 5HKO after Leftovers recovery
As you can see, all of Azumarill's moves don't do that much to this beautiful derp creature.
The first move, recover, is to heal off any damage that is dealt to Quagsire. It can be useful if it does get worn down. Scald and Toxic are filler moves (using Toxic
WILL NOT affect how quickly the Azumarill dies and would be a waste of a turn). Sludge Wave is the most important move here.
132 SpA Quagsire Sludge Wave vs. 92 HP / 0 SpD Azumarill: 140-166 (38.4 - 45.6%) -- guaranteed 3HKO
This probably won't be included because of the 'low' damage done here, however Quagsire will outlast the Azumarill and be able to kill it first. 124 Speed EVs allow it to speed creep the Azumarill, to given it a better chance in this matchup.
Overall, I see Quagsire as a good counter to Azumarill, while not the best one out there, I think it works and would win in a 1v1 against it.