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We consider steady-state heat conduction across a quantum harmonic chain connected to reservoirs modeled by infinite collection of oscillators. The heat, , flowing across the oscillator in a time interval is a stochastic variable and we study the probability distribution function . We compute the exact generating function of at large and the large deviation function. The generating function has a symmetry satisfying the steady-state fluctuation theorem without any quantum corrections. The distribution is non-Gaussian with clear exponential tails. The effect of finite and nonlinearity is considered in the classical limit through Langevin simulations. We also obtain the prediction of quantum heat current fluctuations at low temperatures in clean wires.