In this research, inferential theory for hypothesis testing under general convex cone alternatives for correlated data is developed. While there exists extensive theory for hypothesis testing under smooth cone alternatives with independent observations, extension to correlated data under general convex cone alternatives remains an open problem. This long-pending problem is addressed by (1) establishing that a "generalized quasi-score" statistic is asymptotically equivalent to the squared length of the projection of the standard Gaussian vector onto the convex cone and (2) showing that the asymptotic null distribution of the test statistic is a weighted chi-squared distribution, where the weights are "mixed volumes" of the convex cone and its polar cone. Explicit expressions for these weights are derived using the volume-of-tube formula around a convex manifold in the unit sphere. Furthermore, an asymptotic lower bound is constructed for the power of the generalized quasi-score test under a sequence of local alternatives in the convex cone. Applications to testing under order restricted alternatives for correlated data are illustrated.