We consider a viscous incompressible fluid governed by the Navier-Stokes system written in a domain where a part of the boundary is moving as a damped beam under the action of the fluid. We prove the existence and uniqueness of global strong solutions for the corresponding fluid-structure interaction system in an Lp-Lq setting. The main point in the proof consists in the study of a linear parabolic system coupling the non stationary Stokes system and a damped beam. We show that this linear system possesses the maximal regularity property by proving the R-sectoriality of the corresponding operator. The proof of the main results is then obtained by an appropriate change of variables to handle the free boundary and a fixed point argument to treat the nonlinearities of this system.