A quantum Beilinson algebra is roughly speaking a finite dimensional algebra which is derived equivalent to a quantum projective space.  Since a quantum projective space can be thought of as a noncommutative Fano variety, it is a typical example of a Fano algebra introduced by Minamoto.  Over such an algebra, a notion of regular module was introduced by Herschend, Iyama and Oppermann from the view point of representation theory of finite dimensional algebras.  In this talk, after defining those new notions, we will explicitly calculate moduli spaces of simple regular modules over some low dimensional quantum Beilinson algebras using techniques of noncommutative algebraic geometry.