Hmmh, I'm afraid this is a tricky one ... and I'm also afraid it is again one where multiple answers might qualify as being correct ...
Usually I would have answered that it is the
Baltimore Pratt Street riot on April 18, 1856, when a secessionist mob in Baltimore attacks Massachusetts troops bound for Washington, D.C. Four soldiers and 12 rioters were killed, which made that riot the first blood spilled in the Civil War.
http://www.history.com/this-day-in-history/first-blood-in-the-civil-war
But then I noticed that there is a difference between "the first blood
in the Civil War" and "the first blood
of the Civil War", as it was asked in the question.
The latter could be hinting to a gentleman by the name of
Nicholas Biddle, an African-American soldier passing through Baltimore on April 18, 1861 on his way to Washington and the battlefields of the Civil War.
"On this day in 1861, Nicholas Biddle became “the first blood of the Civil War” in the midst of mob violence in Baltimore."
https://blackthen.com/april-18-1861-nicholas-biddle-first-person-shed-blood-civil-war/
http://www.historynet.com/nicholas-biddle-first-blood.htm (<-- very interesting article, very worth reading!)
However, he is not the only one who is said to be the first one to spill his blood in the Civil War ... he shares that honor with a 17 year old kid from New Hampshire,
Luther C. Ladd:
https://books.google.de/books?id=4g...YAhVBOBQKHcHzAa4Q6AEIUjAH#v=onepage&q&f=false
But then there is even a fourth event that claims to be the first blood of the Civil War, although it had happened years before, a
fight between Member of the House of Representatives Preston Brooks and Senator Charles Sumner , having taken place as early as May 20, 1856:
https://books.google.de/books?id=1C...WMtRQKHTVNC6c4ChDoAQguMAE#v=onepage&q&f=false
The the blows were said to have been so heavy that the cane broke and Sumner was seriously injured.
https://en.wikipedia.org/w/index.php?title=Preston_Brooks&oldid=819680830
I hope that this is enough proof that each of these answers should be regarded as being correct.
But if I have to choose one, I choose the
fight between Preston Brooks and Senator Charles Sumner as my final answer. That one is so unique, I think it qualifies best.