A recurring motif in cellular mechanobiology is the coupled interaction between cell membranes with embedded semiflexible filaments. Their coupled mechanics play a crucial role in the proper functioning of the cell. For instance, in most cells, the membrane surface is embedded in a matrix of protein filaments called the cytoskeletal network, which provides a scaffold for rigidity, force sensing, and actuating shape changes. Another example where the membrane filament interaction is relevant is during cell division, when a ring of actin filaments forms at the cell's equatorial plane. Aided by myosin motors, the ring constricts, leading to the fission of the cell and the formation of two daughter cells. This work presents a computational formulation based on continuum mechanics to study the interaction of fluid membranes embedded with semiflexible filaments. We model the membrane as a thin fluid shell and the filament as a one-dimensional Cosserat continuum. We assume that the filament is tethered to the membrane surface while also being allowed to float freely on it. The fluidity of the membrane and the novel filament-membrane coupling pose unique computational challenges, which are addressed in this work.