We propose a computationally efficient method to construct nonparametric, heteroskedastic prediction bands for uncertainty quantification, with or without any user-specified predictive model. The data-adaptive prediction band is universally applicable with minimal distributional assumptions, with provable strong coverage properties, and easy to implement using standard convex programs. Our approach can be viewed as a novel variance interpolation with confidence and further leverages techniques from semi-definite programming and sum-of-squares optimization. Theoretical and numerical performances for the proposed approach for uncertainty quantification are analyzed.