  The $p$-partite Ramsey number for quadrilateral, denoted by $r_p(C_4,k)$, is
the least positive integer $n$ such that any coloring of the edges of a
complete $p$-partite graph with $n$ vertices in each partition with $k$ colors
will result in a monochromatic copy of $C_4$. In this paper, we present an
upper bound for $r_p(C_4,k)$ and the exact values of $r_p(C_4,2)$ for all
$p\geq2$. In tripartite case we show that $r_3(C_4,k) \leq \lfloor
(k+1)^2/2\rfloor-1$ and the exact value of 4-color tripartite Ramsey number
$r_3(C_4,4)=11$.
