Gamma =
First, you need to recognize MATLAB is case sensitive. Your code does not run, instead, it fails in several places. I fixed it, to at least attempt to run.
If you define the variable Integrand, then tryign to use integrand will be an issue. And the MATLAB function simplify needs to be called simplify. Simplify just gets MATLAB confused.
syms k_x k_y T d t real
Gamma = 1 + 4*cos(k_x)*cos(k_y/sqrt(3)) + 4*cos(k_y/sqrt(3))^2
Integrand = exp(- sqrt(d^2 + t^2*Gamma)/T)
I = int(int(Integrand, k_x, 0, pi), k_y, 0, pi)
simplify(I)
Next, when MATLAB just returns an integral, shown like this, it means it was unable to resolve the integration.
Not every expression you decide to integrate will have an analytical solution, and even if one exist does not mean MATLAB will always find it. In at least the first case, MATLAB failed to return a solution. In fact, the vast majority of things you can write down will have no solution, since you can write down things that are arbitrarily complicated. Anyway, I'm not at all surprised that int failed here.
In some cases, you may be able to do a substitiution or other operation to make the problem more tractable. The human brain can often see things a computer does not, even though the computer is a stubborn beast and will try anything.
When all else fails, you can substitute numerical values for those parameters, and perform a numerical integration, though this is often unsatisfactory to those who want a nice pretty closed form solution. Or you can try other tricks, for example a series approximation to the integrand, etc.
