Sarge's Tale

This is still obtainable, however if you are a brand new HS player, you may be faced with a different experience than what the rest of the comments state. Here's what you need to do:

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• You'll be put into the Jaina tutorial, click skip in the top right once you can to leave the tutorial.
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• Click Modes on the main menu, you will now either see four badge icons, with the fourth being Mercenaries or if you dont see this and instead see Solo Adventure, you will see a list on the right with Book of Mercenaries in this list
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• If you see the four badges, click Mercenaries and complete the tutorial
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• If you see The Book of Mercenaries however, go back to the main menu, click escape, click options, click miscellaneous and click "Skip Apprenticeship" and confirm. You have to be on the main menu to see the Misc option
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• Now click on Modes again and you should see the four badges, with the fourth being Mercenaries. This will save you a ton of time, as the apprenticeship forces you to unlock all the classes.
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• Complete the Mercenaries tutorial up to the point where you unlock the Barrens Bounty, select Quilboar and you should see the mount award pop up after you've loaded into this mission
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If you want a less vague answer for how much of Hearthstone you need to play you need to do Mercenaries mode until you start the Quillboar mission. As soon as you start it and load into the first fight the mount is added to your collection and you can safely Alt F4 Hearthstone.

No, it doesn't fly