This paper deals with the initial-boundary value problem of the biharmonic cubic nonlinear Schr\"odinger equation in a quarter plane with inhomogeneous Dirichlet-Neumann boundary data. We prove local well-posedness in the low regularity Sobolev spaces introducing Duhamel boundary forcing operator associated to the linear equation to construct solutions on the whole line. With this in hands, the energy and nonlinear estimates allow us to apply Fourier restriction method, introduced by J. Bourgain, to get the main result of the article. Additionally, adaptations of this approach for the biharmonic cubic nonlinear Schr\"odinger equation on star graphs are also discussed.