In this paper, we determine the exact value of the $2$-edge-coloring Ramsey number of a connected $4$-clique matching $c(nK_4)$, which is a set of connected graphs containing an $nK_4$ is $13n-3$ for any positive integer $n \geq 3$. This is an extension of the result by Roberts (2017), which is proved only for $n\geq 18$. We also show that the result still holds when $n=2$ provided that $R_2(2K_4) \leq 23$.