Veldkamp polygons are certain graphs $Γ=(V,E)$ such that for each $v\in V$, $Γ_v$ is endowed with a symmetric anti-reflexive relation $\equiv_v$. These relations are all trivial if and only if $Γ$ is a thick generalized polygon. A Veldkamp polygon is called flat if no two vertices have the same set of vertices that are opposite in a natural sense. We explore the connection between Veldkamp quadrangles and polar spaces. Using this connection, we give the complete classification of flat Veldkamp quadrangles in which some but not all of the relations $\equiv_v$ are trivial.