It may be that the integral does not exist. If so, you can still obtain numerical results:
int[Delta_, J_] :=
NIntegrate[
Sqrt[(2*Delta*J*(3 - Cos[Pi*x] - Cos[Pi*y] - Cos[Pi*z]) +
J^2*(3 - Cos[Pi*x] - Cos[Pi*y] - Cos[Pi*z])^2)], {x, 0, 1}, {y,
0, 1}, {z, 0, 1}]
int[1, 1]
(* 3.853312214951417` *)
I used your code as written, but it is usually best to begin user-defined variables in lower case to prevent colliding with reserved words.
Kind regards,
David