Please show us your proof the optimization problem is convex. You haven’t shown the whole problem, so readers can’t know. However, what you do show is a linear fractional form. Is the denominator always positive?
Perhaps this can be handled by techniques along the lines of section 4.3.2 “Linear-fractional programming” of Convex Optimization – Boyd and Vandenberghe, or the problem is quasi-convex and can be handled by the bisection algorithm in section 4.2.5 “Quasiconvex optimization” of that book.
However, perhaps you are not telling us everything? If Q and t were the only optimization variables, and everything else was input data, I believe you would have gotten the error message, {affine}/{affine}, not {convex}/{convex}. So that may put my teh applicability of my preceding paragraph into greater doubt.