Inferring the values and uncertainties of cosmological parameters in a cosmology model is of paramount importance for modern cosmic observations. In this paper, we apply simulation-based inference (SBI) approach to estimate cosmological constraints from a simplified galaxy cluster observation analysis. Using data generated from the Quijote simulation suite and analytical models, we train a machine learning algorithm to learn the probability function between cosmological parameters and the possible galaxy cluster observables. The posterior distribution of the cosmological parameters at a given observation is then obtained by sampling the predictions from the trained algorithm. Our results show that the SBI method can successfully recover the truth values of the cosmological parameters within the 2σ limit for this simplified galaxy cluster analysis, and results in similar posterior constraints obtained with a likelihood-based Markov Chain Monte Carlo method, the current state-of the-art method used in similar cosmological studies.