Nidoking is usually better suited for your competitive needs because his higher Speed and offenses, in addition to Megahorn, make it a lot easier to pulverize opposing teams. However, Nidoqueen is much better at using Counter due to higher sturdiness. Because of this tankish nature, Nidoqueen is also better at using the nice resistances given to the Poison / Ground typing, but still you should be wary of these nasty weaknesses to Earthquake, Psychic, Ice Beam, and Surf.
The standard Nidoking set, applied to Nidoqueen. Nidoqueen lacks Megahorn, but it can remedy that by hitting Meganium with STAB Sludge Bomb and Solrock with Shadow Ball. Ice Beam OHKOs Gligar and does a lot of damage to other Grounds, as well as Fearow, who is immune to two of your moves.
Requires prediction to work. Your main attacking move here is Earthquake, so like with above set, hit Fliers like Gligar with Ice Beam, Shadow Ball is for Solrock, and Sludge Bomb is for Meganium (and it hits Fearow quite hard as well). This set makes Nidoqueen do notable damage if used right.
If you're using Ice Beam for Gligar, use 56 Special Attack EVs and Hasty or Naive nature (taking EVs from HP and Speed), so you can OHKO it.
Counter is an awesome move on Nidoqueen, since she usually survives even STAB Earthquakes if she has a defensive EV spread. Charm can be annoying as hell for physical switch-ins like Fearow, Snorlax or any Ground. Roar Nidoqueen can be used for teams based around Baton Pass and it can easily Roar away Ninjask or Scizor as well. Nidoqueen is defensive enough to pull off a Rest / Sleep Talk set, but it's better to use her outstanding versatility. Substitute helps against Weezing if it's not running Flamethrower or Fire Blast.
Hidden Power Grass 2HKOes SwaQuagsire and Whiscash, but you should be using it on the switch or they will KO you with Earthquake, Surf or Hydro Pump. Use Crunch if you're using a Nidoqueen that's focused on Special Attack (though that's not recommended as it lacks STAB); it will damage Solrock and Misdreavus badly.
If you want to OHKO max HP Gligar with Ice Beam guaranteed, you need 200 Special Attack. The main sets should have offensive spreads like suggested, but if you're using Counter or just want a more sturdy Nidoqueen, I recommend using 252 HP / 140 Attack / 36 Defense / 80 Speed with Adamant nature (when not using any special attacks). With this, you survive STAB Earthquakes from the likes of Sandslash. This means you'll be easily living through Nidoking, non-Swords Danced Gligar and any non-STAB Earthquake.
Checks and Counters
Nidoqueen is versatile, but since its attacking stats are on the low side it doesn't hurt much if it's not dealing STAB or super effective damage.
It's mostly Waters and Grounds that Nidoqueen fears. Politoed, Slowking, and Blastoise take on non-Choice Banded Nidoqueen easily, though they may get worn down after a while. If Nidoqueen is not using Hidden Power Grass, Quagsire and Whiscash are failsafe counters in the same way. Poliwrath usually doesn't run Earthquake or a Water move, but if it does it's a good Nidoqueen counter as well.
Sandslash takes Ice Beam even worse than Donphan does, so it's only good as an early game counter that shouldn't switch into Ice Beam, and preferably not into Earthquakes either. Watch out for Sludge Bomb's Poison chance, too, as well as the rare Hidden Power Grass. Gligar can come in with some prediction to avoid Ice Beam, but it cannot OHKO Nidoqueen with Earthquake when Nidoqueen is at (near) full health, so just use it as a finisher. It will get OHKOed by most Ice Beaming Nidoqueen. Solrock, Misdreavus and Lunatone are immune to Earthquake and resist Sludge Bomb, so as long as they don't eat a Shadow Ball they're good counters. Solrock can even set up a Reflect to avoid too much damage. Solrock can Earthquake Nidoqueen (watch out for Counter), Misdreavus can Perish Trap or Calm Mind up, Lunatone can Calm Mind up and use Ice Beam / Psychic.