In this paper we introduce the concept of equilibrium for a non-cooperative multiobjective bimatrix game with payoff matrices and goals of Z-Numbers. In the recent studies of the authors, the problem of finding equilibrium for a non-cooperative bimatrix of Z-Numbers are investigated. Multiple payoffs are often dealt with in games because a decision making problem under conflict usually involves multiple objectives or attributes such as cost, time and productivity. We let each of the objectives of the problem correspond to each of the payoffs of the game. To aggregate multiple goals, we employ two basic methods, one by weighting coefficients and the other by a minimum component. In order to find the equilibrium solution in such circumstances, we developed a mathematical programming problem to maximize the aggregated goal subject to constraint of satisfying an aspiration level of confidence in the equilibrium solution. Finally a method is presented to determine the equilibrium solution with respect to the level of achievement to the aggregated goal.