You're a Mean One...

As of 11.0.7, there is a new Express Transportation NPC called Father Winter's Helper that will take you on a sleigh ride to the Greench's location (you port directly once the sleigh leaves Ironforge or Orgrimmar).

Ironforge /way 31.7, 73.1
Orgrimmar /way 50.3, 79.5

Note: this tutorial is for Alliance version of the quest. Horde version is here.

Alliance players can get to the quest location in following ways:

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• Fly directly from Ironforge. This is slow and you risk dismounting near Arathi Highlands, so I usually avoid this option.
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• Direct portal from Legion Dalaran to old Dalaran location in Hillsbrad Foothills was very handy in 7.x, but starting from 8.0 it is open only if you are on relevant quest. Use it if you can access it, just make sure you use some kind of slow fall, as you will appear in the midair.
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• Mages can use Ancient Teleport: Dalaran or Ancient Portal: Dalaran to get to old Dalaran location too. Also remember to use some slowfall, or use teleport when on flying mount.
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• When Alliance controls Arathi Highlands, you can use portal in Boralus to get there. You need to have completed BfA intro quests up to A Nation Divided to see it.
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• If you completed Stratholme pet dungeon, Manapoof can teleport you to its entrance in Eastern Plagueland. Its similar distance as from Ironforge, but you do not risk dismounting during flight.
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• Closest Hearthstone location is at Wildhammer Keep in Hinterlands (at Aerie Peak). I recommend going there when you complete this quest for the first time and set your HS, so you could get there faster next time.
  TomTom: /way The Hinterlands 14.15 44.75 Wildhammer Keep
  
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• Dark Iron Dwarves can use their mole machine to get to Aerie Peek, but personally I recommend to use it for trip back to Ironforge.
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• Druids can use the Dreamway to Hinterlands (thanks Flanell for hint). Portal leads to east side of Hinterlands, Seradane.
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Also remember that you can get this quest from [npc=13636] in Hillsbrad Foothills, so you do not have to start from Ironforge.

You can return to the Ironforge in following ways:

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• Fly directly to Ironforge - slow and you risk dismounting.
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• Mages can use portal or teleport to Ironforge. If you are not a mage you can ask everyone for portal after the battle, some friendly mage player may open one for you.
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• You can set your Hearthstone to The Stonefire Tavern in Ironforge if you want.
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• Teleport yourself to Stormwind, then use portal to Boralus, and from there use portal to Ironforge. You can get to SW using guild cloaks (Cloak of Coordination, Wrap of Unity, Shroud of Cooperation). You can also use your HS to get to some main city (e.g. Valdrakken) and use Stormwind portal there. Also remember about Dalaran Hearthstone and Garrison Hearthstone if you unlocked them.
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• Dark Iron Dwarves can use their mole machine to get to IronForge - it will emerge close to the city gate.
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• Druids can use the Dreamway to Duskwood, then fly from there to Stormwind and use portals as described above.
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Merry Christmas to Everyone!

This quest has been changed as of 4.3. It now appears to be a Daily Group quest, requiring lvl 80 to pick it up. Also, it rewards [item=73792]. Perhaps Blizzard made the Greench into an Un-Instanced Holiday Boss rather than have a LFD thing like previous Holidays.

Edit:[item=73792] will have a chance to contain [item=73797], some Justice Points if you are not capped, as well as other assorted holiday pets and toys.