“One day the great European War [will] come out of some damned foolish thing in the Balkans.”
http://izquotes.com/quote/303800
The only source I've found says it was attributed BY Churchill in 1923.
How likely is the quote to be authentic?
As is almost always the way with (verbal) quotations, tracing exact provenance and establishing truth is an uphill battle. So take this with a pinch of salt.
Your quote appears to be a paraphrase of this:
Europe today is a powder keg and the leaders are like men smoking in an arsenal ... A single spark will set off an explosion that will consume us all ... I cannot tell you when that explosion will occur, but I can tell you where ... Some damned foolish thing in the Balkans will set it off.
That's (allegedly) Bismarck speaking during the Congress of Berlin in 1878. It's the origin of the 'Powder Keg of Europe' trope about the Balkans, and it's one of those quotations that has been very widely paraphrased and parroted.
It could be apocryphal, or it could not. Hard to say, really.