∂z/∂x=-(q-y) is the gradient of the saddle-shaped surface in the x direction.
∂z/∂y=-(p-x) is the gradient of the surface in the y direction.
It's not clear what's meant by "point x-axis". If the point x=p is implied, the partial derivatives become:
∂z/∂y=-(p-x)=0 when x=p which means that the gradient is zero, that is, there is a turning point of the surface because the tangent at the point x=p is parallel to the y-axis.
∂z/∂x=-(q-y)=y-q. When y=q (the point (p,q,p2+q2)) the gradient is zero (the tangent is parallel to the x-axis); but otherwise the gradient depends on whether y is greater than or less than q. Note that the point (p,q,p2+q2) is a saddle-point on the surface (the lowest point on the saddle-like surface).
The question needs clarification.