To solve the problem generated by unknown Lipschitz constants for the gradient of the smooth part of the objective function, it proposes a new backtracking strategy for composite multi-objective optimization problems based on the accelerated proximal gradient algorithm. This strategy constructs an update rule for an estimated sequence of Lipschitz constants that satisfies a specific equality relationship, allowing the sequence to be updated in a non-monotonic way. Under appropriate conditions, it is proven that all cluster points of the sequence generated by the algorithm are weakly Pareto optimal solutions. Furthermore, the sublinear convergence rate O(1/k2) of the algorithm is established using merit functions. Numerical experiments demonstrate that, compared to the accelerated proximal gradient algorithm without the backtracking strategy, the proposed algorithm exhibits significant advantages regarding runtime, iteration count, and function evaluation count.