In this note, we develop a numerical approach to the problem of optimal stopping of discrete-time continuous-state partially observable Markov processes (POMPs). Our motivation is to find approximate solutions that provide lower and upper bounds on the value function such that the gap between the bounds can provide a practical measure of the quality of the solutions. To this end, we develop a filtering-based duality approach, which relies on the martingale duality formulation of the optimal stopping problem and the particle filtering technique. We show that this approach complements an asymptotic lower bound derived from a suboptimal stopping time with an asymptotic upper bound on the value function. We carry out error analysis and illustrate the effectiveness of our method on an example of pricing American options under partial observation of stochastic volatility.