What makes you think volume flux for an incompressible fluid isn't zero?
Because it is.
Actually, I'd like to think of it the other way around.
Suppose we take a small volume at a fixed point.
And suppose we can say that at any moment in time the amount fluid that goes in is equal to the amount that goes out.
Then that means 2 things:
1. The fluid is incompressible, or rather it is never compressed or decompressed.
2. The divergence is zero (as a result of the definition of divergence).
As for the definition of divergence, in words it is (abbreviated from wikipedia):
"the divergence represents the outward flux from an infinitesimal volume"
Or more formally (also from wikipedia):
"the divergence of a vector field F at a point p is defined as the limit of the net flow of F across the smooth boundary of a three dimensional region V divided by the volume of V as V shrinks to p."