This vector field has all vectors pointing toward x-axis, and the length of vectors is only a function of y. So the field will appear very uniform. You also need to consider the x and y grid to plot it properly. Please try the following code.
[x,y,z] = meshgrid(linspace(0,0.5,10), linspace(0,2,100), linspace(0,0.5,10));
% We now define our 3 vector−field functions
P = y.^2-2*y;
Q = 0*y;
R = 0*y;
% Plot the 3D vector field with quiver3
figure(2)
quiver3(x,y,z,P,Q,R)
grid on
ylim([0,2])
xlabel('x')
ylabel('y')
zlabel('z')