We introduce the notion of the weak Brehmer's condition and prove that the Cauchy transform for a representation of a right-angled Artin monoid is bounded under such conditions. As a result, we obtain the Poisson transform and $*$-regular dilation for a family of operators that satisfies the weak Brehmer's condition and the property (P). This generalizes Popescu's notion of Cauchy and Poisson transforms for commuting families of row contractions.