This letter proposes a novel robust interval observer for a two-dimensional (treated as a synonym for a double-indexed system) linear time-invariant discrete-time system described by the Fornasini-Marchesini second model. This system is subject to unknown but bounded state disturbances and measurement noise. Built on recent interval estimation strategies designed for one-dimensional systems, the proposed observer is based on the introduction of weighting matrices which provide additional degrees of freedom in comparison with the classical structure relying on a change of coordinates. Linear matrix inequality conditions for the exponential stability and peak-to-peak performance of a two-dimensional system described by the Fornasini-Marchesini second model are then proposed, and applied to the design of a robust interval observer. Numerical simulation results are provided to show the efficiency of the proposed estimation strategy.