Shoot the crow is an English idiom meaning 'to leave',
in England and Scotland, or, in America, 'to leave without paying for a drink'.
While there are any number of websites willing to tell you that this is rhyming slang (shoot the crow, go), or Regency slang, or a number of even more random claims, it is first recorded in writing in 1844, in a Scottish poem by John Ramsay.1
"Dear Britain, should it be thy hap
Again to face another Nap
Just lea' thae chiels to tent the crap,
And shoot the craw
The Scots version, naturally, uses the Scots craw rather than crow, but this seems to be the first written use of the idiom meaning 'to leave'. It's hard to be certain, as it is also possible that he is saying that you should let the children chase off crows while tending the crops; unfortunately, 'shoot' has enough meanings in Scots to be thoroughly ambiguous.
The phrase thereafter disappeared from published works for decades, indicating that it circulated primarily in informal, everyday speech; we have no real evidence as to when it originated or how common it was. It is still used today, in both Scots and English forms, although it is hardly common. It would appear that whenever it originated, 'shoot' in this case means 'shove off' or 'pitch out', rather than 'fire at'.
Meanwhile, in 1890 in an English dictionary of slang2 collecting a miscellany of phrases (with no attempt to give origins) defined 'shoot the crow' as meaning to 'run off without paying'. The next appearance, in 1897 -- and in another slang dictionary3 -- credits it to the American west. This volume is quite specific: in America it refers to leaving a bar without paying the barkeep for a drink, and is probably a reference to "crow-whiskey", meaning high-quality whiskey.
Crow-whiskey, in this case, presumably refers to 'Old Crow Whiskey'. While Old Crow Whiskey is today known as a comparatively cheap brand of whiskey, originally it had a reputation as a quality brand, and was often referred to simply as 'Crow', or if well-aged, 'Old Crow'. It was developed by James C. Crow circa the 1830s, so the time-frame fits.
Shoot, meanwhile, would be used as a variant of 'moving quickly', rather than in a reference to firearms or shoving off. Today we would probably say chug. It is also possible that it might be a mis-transcription of chute.
There is almost certainly some connection between the two sayings, with the Americans repurposing a foreign phrase to fill a linguistic need, perhaps with some false etymology made up to satisfy the curious. These days, when this phrase is used at all, it is most often used in the original, Scottish sense.
Notes:
1. Woodnotes of a Wanderer (1844), by John Ramsay, specifically in the poem The Loudon Campaign.
2. Slang and Its Analogues Past and Present (Volume 6), By William Ernest Henley, 1890: to run off without paying, to bilk.
3. A Dictionary of Slang, Jargon & Cant Embracing English, American, and Anglo-Indian Slang, Pidgin English, Gypsies' Jargon and Other Irregular Phraseology (Volume 2), Editors Albert Barrère and Charles Godfrey Leland, 1897.