That really depends on what you mean when you talk about whiskey. The art of distillation of spirits spread pretty quickly after it's introduction to Europe from the Middle East in the 13th century arriving in Ireland and Scotland by at least the middle 15th century if not earlier. So the most technical answer would be that the first whiskeys were distilled in Ireland and later Scotland by monks using their most prevalent source of local sugar which would have been malted barley in the 15th century.

However, the catch is that these early distillates are not anything that you would recognize today as whiskey because undoubtedly when you think of whiskeys you are thinking of a brown, aged spirit containing 40%-50% alcohol by volume whereas these earlier spirits were much more aggressive. They were of a significantly higher proof (typically 70%+) and the process of finishing the distillate in oak barrels had not been developed yet so you would probably more likely call this moonshine whereas the earliest European distillers referred to it as aqua vitae, the water of life. It wasn't until the late 17th century that you really see the evolution of this spirit into modern whiskey with the addition of water in order to control the proofing and later in the 18th century with the finishing of the whiskeys in French oak sherry casks. After these innovations larger scale distilleries started to open very rapidly including many still open today both in Scotland (Glenturret - 1775, Bowmore - 1779, Oban - 1794) and in Ireland (Kilbeggan - 1757, Bushmills - 1784).