Using convolutional neural networks, deep learning-based reduced-order models have demonstrated great potential in accelerating the simulations of coupled fluid-structure systems for downstream optimization and control tasks. However, these networks have to operate on a uniform Cartesian grid due to the inherent restriction of convolutions, leading to difficulties in extracting fine physical details along a fluid-structure interface without excessive computational burden. In this work, we present a quasi-monolithic graph neural network framework for the reduced-order modelling of fluid-structure interaction systems. With the aid of an arbitrary Lagrangian-Eulerian formulation, the mesh and fluid states are evolved temporally with two sub-networks. The movement of the mesh is reduced to the evolution of several coefficients via proper orthogonal decomposition, and these coefficients are propagated through time via a multi-layer perceptron. A graph neural network is employed to predict the evolution of the fluid state based on the state of the whole system. The structural state is implicitly modelled by the movement of the mesh on the fluid-structure boundary; hence it makes the proposed data-driven methodology quasi-monolithic. The effectiveness of the proposed quasi-monolithic graph neural network architecture is assessed on a prototypical fluid-structure system of the flow around an elastically-mounted cylinder. We use the full-order flow snapshots and displacements as target physical data to learn and infer coupled fluid-structure dynamics. The proposed framework tracks the interface description and provides the state predictions during roll-out with acceptable accuracy. We also directly extract the lift and drag forces from the predicted fluid and mesh states, in contrast to existing convolution-based architectures. The proposed reduced-order model via graph neural network has implications for the development of physics-based digital twins concerning moving boundaries and fluid-structure interactions.