Let's assume that, per Rob Pratt's comment, your $Q$ is symmetric and positive definite. Your approximation is correctly derived, and you can certainly try it. Due to convexity, the linear approximation will be conservative, so you may get values of $z$ that are smaller than they should be, leading to incorrect (suboptimal) solutions.
Linearizing the product exactly would involve a bit over a half million new continuous variables and two to three times that many new constraints. The added constraints would mercifully be sparse. Depending on the dimensions of the rest of your problem (and your computational horsepower and patience), it's quite possible that a good quality MIP solver could handle the linearized version.
Most current commercial solvers (and at least some open source solvers) can handle convex quadratic objectives and second order cone constraints, so you could solve the problem with no modifications or with the constraint $x' Q x - z \le 0,$ without the tangential approximation.