An $L(j,k)$-labeling of a graph G is a vertex labeling such that the difference of the labels of any adjacent vertices is at least $j$ and that of any vertices of distance two is at least $k$ for given $j$ and $k$. The minimum span of all L(2, 1)-labelings of G is called the ${lambda}$-number of G and is denoted by ${lambda}(G)$. In this paper, we find a lower bound of the ${lambda}$-number of the Cartesian product $K_m{Box}C_n$ of the complete graph $K_m$ of order $m$ and the cycle $C_n$ of order $n$. In fact, we show that when $n{geq}3$, ${lambda}(K_4{Box}C_n){geq}7$ and the equality holds if and only if n is a multiple of 8. Moreover when $m{geq}5$, ${lambda}(K_m{Box}C_n){geq}2m-1$ and the equality holds if and only if $n$ is even.