First of all, I haven't seen the show myself and this answer is only based on Wikipedia combined with common sense (resummarizing my comments into a proper answer).
An interesting aspect is that the Ichabod Crane from the TV-show does in fact come originally from the 18th century. So the only reasonable possibility to combine the TV-show with the original story would be that Irving actually based his story on this same real Ichabod Crane and his adventures in the past. But I think in this case the writers of the TV-show would have adressed this aspect at one point or another and if not a single character in the TV-show ever adresses this point or seems to ever have heard about the traditional story, you can safely assume it doesn't exist for them.
Another way to view the TV-show would be like Wikipedia does, as "a modern-day retelling of the 1820 short story" and thus something like a remake. Thus this TV-show is the one and only Sleepy Hollow story and Irving's wouldn't exist in this universe, as people in stories don't hear their own stories, which would need them to be aware of being fictional. This is IMHO strengthened by Wikipedia's plot summary saying:
In 1781, Ichabod Crane "dies" in the midst of the Revolutionary War
while on a mission for General George Washington. He rises from his
"grave" in modern day Sleepy Hollow after the Headless Horseman,
(revealed to be Death, one of the Four Horsemen of the Apocalypse), is
summoned back from his watery grave by an unknown party. The
resurrection of the one causes the resurrection of the other due to
the intertwining of their fates as a result of their blood mixing
shortly after Crane decapitated the Headless Horseman on the field of
So Crane's past history with the Horseman isn't similar to the original story and only given to provide some relationship to him, and Crane's anachronicity is done to provide an interesting out-of-time aspect to the show (and to carry over the "early U.S./colonial history"-aspect of the orignal story to some degree) rather than to hint at any actual relation of him to the Irving story.