Step 1 :First, we find the asymptotes of the given functions.
Step 2 :For the function $y = b^{x-1} + 2$, the horizontal asymptote is $y = 2$.
Step 3 :For the function $y = \log_b(x-1) + 2$, the vertical asymptote is $x = 1$.
Step 4 :Thus, the intersection point of the asymptotes is $(1, 2)$.
Step 5 :\(\boxed{(1, 2)}\)