The blockchain technology has been rapidly popularizing and gaining a quick mileage in many domains owing to its innate nature, so there must be several tremendous market opportunities and uncertainties for the blockchain cloud service. We propose a Cournot duopoly game in a blockchain cloud services market for cyber-physical-social systems (CPSS). Introducing Brownian noise disturbance to the deterministic Cournot duopoly game, we construct a stochastic Cournot duopoly game in a blockchain cloud service market with uncertainty. We analyze the stochastic stability of the Cournot duopoly game by the Lyapunov exponent method, the Lyapunov function, and the singular boundary theory. Then we investigate the stochastic Hopf bifurcation of the Cournot duopoly game by the invariant measure theory. At last, we numerically validate these results by the probability density. These results of stability and Hopf bifurcation illustrate complex behaviors of the stochastic Cournot duopoly game in the blockchain cloud services market.