I noticed on a recent trip to Austria, Germany and Italy that in the
Alps, the rivers are bluish green in color and very, very clear.  The
rocks appear to be white which leads me to believe there is chlorine
in the water that has bleached them.  What does cause the beautiful
color and how does it get into the streams?

Natural limestone and sulfuric acid combine in the stream and cause
the effect you seen on the rocks and the clear water...