Uncertain multiobjective optimization problems arise in various real-world scenarios where ob-jectives are affected by uncertainty. To address this, we propose a quasi-Newton method to solve the robust counterpart of an uncertain multiobjective optimization problem under an arbitrary  finite uncertainty set. The robust counterpart is formulated as a nonsmooth deterministic multiobjective optimization problem, where we construct a sub-problem using Hessian approximation to determine a descent direction.     An    Armijo-type inexact line search technique is introduced to compute an appropriate step length, and a modified BFGS formula ensures positive definiteness of the Hessian matrix at each iteration. By incorporating these components, we develop a quasi-Newton descent algorithm for the robust counterpart and establish its convergence under standard assumptions, proving a superlinear convergence rate. Numerical experiments validate the e effectiveness of our method by comparing it with the weighted sum method through a performance profile, demonstrating its efficiency and robustness in solving uncertain multiobjective problems.