For groups $A,B,C$ and $D$ is it true that
$${\rm Hom}(A\times B,C\times D)\cong{\rm Hom}(A, C)\times{\rm Hom}(A, D)\times{\rm Hom}(B, C)\times{\rm Hom}(B, D),$$
where $\times$ is external direct product. I am searching results related to this. Please suggest to me any book. Thank you.
No. If $A,B,C$ are groups, then we have $$\hom(C,A \times B) \cong \hom(C,A) \times \hom(C,B),$$ but $$\hom(A \times B,C) \cong \{(f,g) \in \hom(A,C) \times \hom(B,C) : f,g \text{ commute elementwise}\}.$$ here, $f : A \to C$, $g :B \to C$ commute elementwise if for all $a \in A$ and all $b \in B$ we have $$f(a) g(b) = g(b) f(a).$$ If $C$ is abelian, this is automatic, so that $ \hom(A \times B,C) \cong \hom(A,C) \times \hom(B,C)$ in that case.
Combining these results gives that $\hom(A \times B,C \times D)$ is isomorphic to the subset of $\hom(A,C) \times \hom(A,D) \times \hom(B,C) \times \hom(B,D)$ of those $(f,g,u,v)$ such that $(f,g) : A \to C \times D$ commutes elementwise with $(u,v) : B \to C \times D$, which means that $f$ commutes elementwise with $u$ and $g$ commutes elementwise with $v$.
As a book recommendation, I would suggest Aluffi's Algebra: Chapter 0.