Approximate Bayesian computation (ABC) is used in this study to estimate unknown model parameter values, as well as uncertainties, in a nonequilibrium anisotropy closure for Reynolds averaged Navier-Stokes (RANS) simulations. The ABC approach does not require the direct computation of a likelihood function, thereby enabling substantially faster estimation of un-known parameters as compared to full Bayesian analyses. The approach also naturally provides uncertainties in parameter estimates, avoiding the artificial certainty implied by optimization methods for determining unknown parameters. Details of the ABC approach are described, including the use of a Markov chain Monte Carlo technique to accelerate the parameter estimation, and unknown model parameters are estimated based on turbulence kinetic energy reference data for four impulsively sheared homogeneous turbulence test cases, as well as periodically sheared homogeneous turbulence for five different shearing frequencies. The ABC method is shown to yield parameter values for the nonequilibrium anisotropy closure that provide good agreement between model results and the reference data.