The question is about cycle-types types of elements of the wreath product $G = (S_n \times S_n) \ltimes C_2 \cong S_n \wr C_2$ in its product action on $\{1,\ldots, n\} \times \{1,\ldots, n\}$.
The permutation $(\sigma, \rho, c) \in G$ is conjugate, by $(\rho^{-1},\mathrm{id}_{S_n},1)$, to $g = (\rho\sigma, \mathrm{id}_{S_n}, c)$. Suppose
$$\rho\sigma = (x_1,\ldots, x_a)(y_1,\ldots, y_b) \ldots .$$
Let $X = \{x_1,\ldots, x_a\}$ and $Y = \{y_1,\ldots, y_b\}$. Take all $x$ indices modulo $a$ and all $y$ indices modulo $b$. With this convention, $(x_i,y_j)g = (y_j,x_{i+1})$ and $(y_j,x_i)g = (x_i,y_{j+1})$. Hence
$$(x_i,y_j)g^2 = (x_{i+1},y_{j+1})$$
and so $(x_1,y_1)$ is in a cycle of $g^2$ of length $\mathrm{lcm}(a,b)$. Since $g$ swaps the disjoint sets $X \times Y$ and $Y \times X$, the cycle of $g$ containing $(x_1,y_1)$ has length $2\mathrm{lcm}(a,b)$. Therefore $g$ acts on $X \times Y \cup Y \times X$ with cycle type
$$\bigl((2\mathrm{lcm}(a,b))^{ab/\mathrm{lcm}(a,b)}\bigr).$$
For the action on $X \times X$ things are slightly fiddlier: if $a$ is even then there are $a/2$ cycles each of length $2a$; if $a$ is odd then the cycle type is
$$\bigl( (2a)^{(a-1)/2}, a \bigr).$$
Putting these together determines the cycle type of $g$ and hence the cycle index of $G$.