Uncertainty quantification (UQ) is an emerging field that focuses on characterizing, quantifying, and potentially reducing, the uncertainties associated with computer simulation models used in a wide range of applications. Although it has been successfully applied to computer simulation models in areas such as structural engineering, climate forecasting, and medical sciences, this powerful research area is still lagging behind in materials simulation models. These are broadly defined as physics-based predictive models developed to predict material behavior, i.e., processing-microstructure-property relations and have recently received considerable interest with the advent of emerging concepts such as Integrated Computational Materials Engineering (ICME). The need of effective tools for quantifying the uncertainties associated with materials simulation models has been identified as a high priority research area in most recent roadmapping efforts in the field. In this paper, we present one of the first efforts in conducting systematic UQ of a physics-based materials simulation model used for predicting the evolution of precipitates in advanced nickel–titanium shape-memory alloys (SMAs) subject to heat treatment. Specifically, a Bayesian calibration approach is used to conduct calibration of the precipitation model using a synthesis of experimental and computer simulation data. We focus on constructing a Gaussian process-based surrogate modeling approach for achieving this task, and then benchmark the predictive accuracy of the calibrated model with that of the model calibrated using traditional Markov chain Monte Carlo (MCMC) methods.