  In a $(G^1,G^2)$ coloring of a graph $G$, every edge of $G$ is in $G^1$ or
$G^2$. For two bipartite graphs $H_1$ and $H_2$, the bipartite Ramsey number
$BR(H_1, H_2)$ is the least integer $b\geq 1$, such that for every $(G^1, G^2)$
coloring of the complete bipartite graph $K_{b,b}$, results in either
$H_1\subseteq G^1$ or $H_2\subseteq G^2$. As another view, for bipartite graphs
$H_1$ and $H_2$ and a positive integer $m$, the $m$-bipartite Ramsey number
$BR_m(H_1, H_2)$ of $H_1$ and $H_2$ is the least integer $n$, such that every
subgraph $G$ of $K_{m,n}$ results in $H_1\subseteq G$ or $H_2\subseteq
\overline{G}$. The size of $m$-bipartite Ramsey number $BR_m(K_{2,2},
K_{2,2})$, the size of $m$-bipartite Ramsey number $BR_m(K_{2,2}, K_{3,3})$ and
the size of $m$-bipartite Ramsey number $BR_m(K_{3,3}, K_{3,3})$ have been
computed in several articles up to now. In this paper we determine the exact
value of $BR_m(K_{2,2}, K_{4,4})$ for each $m\geq 2$.
