We present a quadratic outer approximation scheme for solving general convex integer programs, where suitable quadratic approximations are used to underestimate the objective function instead of classical linear approximations. As a resulting surrogate problem we consider the problem of minimizing a function given as the maximum of finitely many convex quadratic functions having the same Hessian matrix. A fast algorithm for minimizing such functions over integer variables is presented. Our algorithm is based on a fast branch-and-bound approach for convex quadratic integer programming proposed by Buchheim, Caprara and Lodi [Mathematical Programming 135:369-395 (2012)]. The main feature of the latter approach consists in a fast incremental computation of continuous global minima, which are used as lower bounds. We generalize this idea to the case of k convex quadratic functions, implicitly reducing the problem to 2^k-1 convex quadratic integer programs. Each node of the branch-and-bound algorithm can be processed in O(2^kn) time. Experimental results for a class of convex integer problems with exponential objective functions are presented. Compared with Bonmin's outer approximation algorithm B-OA and branch-and-bound algorithm B-BB, running times for both ternary and unbounded instances turn out to be very competitive.