Uncertainty and delayed reactions in human driving behavior lead to stop-and-go traffic congestion on freeways. The freeway traffic dynamics are governed by the Aw-Rascle-Zhang (ARZ) traffic Partial Differential Equation (PDE) models with unknown relaxation time. Motivated by the adaptive traffic control problem, this paper presents a neural operator (NO) based adaptive boundary control design for the coupled 2$\times$2 hyperbolic systems with uncertain spatially varying in-domain coefficients and boundary parameter. In traditional adaptive control for PDEs, solving backstepping kernel online is computationally intensive, as it requires significant resources at each time step to update the estimation of coefficients. To address this challenge, we use operator learning, i.e. DeepONet, to learn the mapping from system parameters to the kernels functions. DeepONet, a class of deep neural networks designed for approximating operators, has shown strong potential for approximating PDE backstepping designs in recent studies. Unlike previous works that focus on approximating single kernel equation associated with the scalar PDE system, we extend this framework to approximate PDE kernels for a class of the first-order coupled 2$\times$2 hyperbolic kernel equations. Our approach demonstrates that DeepONet is nearly two orders of magnitude faster than traditional PDE solvers for generating kernel functions, while maintaining a loss on the order of $10^{-3}$.