The objective of this study was to demonstrate an efficient methodology for the quantification of mixed (aleatory and epistemic) uncertainties in hypersonic flow computations. In particular, the approach was used to quantify the uncertainty in surface heat flux to the spherical non-ablating heat-shield of a reentry vehicle at zero-angle of attack due to epistemic and aleatory uncertainties that may exist in various parameters used in the numerical solution of hypersonic, viscous, laminar blunt-body flows with thermo-chemical non-equilibrium. Three main uncertainty sources were treated in the computational fluid dynamics (CFD) simulations: (1) aleatory uncertainty in the freestream velocity, (2) epistemic uncertainty in the recombination efficiency for a partially catalytic wall boundary condition, and (3) epistemic uncertainty in the binary-collision integrals. Second-Order Probability utilizing a stochastic response surface obtained with quadrature-based nonintrusive polynomial chaos was used for the propagation of mixed uncertainties. The uncertainty quantification approach was validated on a stochastic model problem with mixed uncertainties for the prediction of stagnation point heat transfer with Fay-Riddell relation, which included the comparison with direct Monte Carlo sampling results. In the stochastic CFD problem, the uncertainty in surface heat transfer was obtained in terms of intervals at different probability levels at various locations including the stagnation point and the shoulder region. A non-linear global sensitivity analysis based on Sobol indices showed that the velocity and recombination efficiency were the major contributors to the uncertainty in the stagnation point heat flux for the reentry case considered in this study.