Learning solution operators for systems with complex, varying geometries and parametric physical settings is a central challenge in scientific machine learning. In many-query regimes such as design optimization, control and inverse problems, surrogate modeling must generalize across geometries while allowing flexible evaluation at arbitrary spatial locations. In this work, we propose Arbitrary Geometry-encoded Transformer (ArGEnT), a geometry-aware attention-based architecture for operator learning on arbitrary domains. ArGEnT employs Transformer attention mechanisms to encode geometric information directly from point-cloud representations with three variants-self-attention, cross-attention, and hybrid-attention-that incorporates different strategies for incorporating geometric features. By integrating ArGEnT into DeepONet as the trunk network, we develop a surrogate modeling framework capable of learning operator mappings that depend on both geometric and non-geometric inputs without the need to explicitly parametrize geometry as a branch network input. Evaluation on benchmark problems spanning fluid dynamics, solid mechanics and electrochemical systems, we demonstrate significantly improved prediction accuracy and generalization performance compared with the standard DeepONet and other existing geometry-aware saurrogates. In particular, the cross-attention transformer variant enables accurate geometry-conditioned predictions with reduced reliance on signed distance functions. By combining flexible geometry encoding with operator-learning capabilities, ArGEnT provides a scalable surrogate modeling framework for optimization, uncertainty quantification, and data-driven modeling of complex physical systems.