In this paper we present the derivation and the application of an adaptive nonintrusive spectral technique for uncertainty quantification. Spectral techniques can be used to reconstruct stochastic quantities of interest by means of a Fourier-like expansion. Their application to uncertainty propagation problems can be performed in a nonintrusive fashion by evaluating a set of projection integrals that is used to reconstruct the spectral expansion. We present the derivation of a new adaptive quadrature algorithm, based on the definition of a sparse grid, which can be used to evaluate these spectral coefficients. This new adaptive algorithm is applied to a reference uncertainty quantification problem consisting of a coupled time-dependent model. The benefits of using such an adaptive method are analyzed and discussed from the uncertainty propagation and computational points of view.