It all depends on your definition of "The Civil War". If you define the war as the war on slavery then I would define bleeding Kansas as the first conflict, however unorganised. If you define the war as the war between the confederacy and the Union, then no. Bleeding Kansas was only a catalyst that sparked the civil war. Of course this answer will definitely differ between authors and sources, however I have found it is generally considered that bleeding Kansas is not seen as the starting of the war, and is merely conflicts between settlers about the continuation of Slavery, which was one of the major reasons for the civil war.

Recommended Reading

• Bleeding Kansas: Contested Liberty in the Civil War Era by Nicole Etcheson

• The Start of the Civil War: The Secession of the South, Fort Sumter, and First Bull Run (First Manassas) by Charles River

Obviously these wont give you all the answers, however they are good for some context.