Let $S(x) = S_q(x)$. Sub $n = q^k + r$ where $k$ is some large number and $r < \frac{q^k}{a} - \max \{ b,b' \}$. Then we have $S(an+b) = S(aq^k + ar + b) = S(aq^k) + S(ar + b)$(because the $ar+b$ is too small to interfere with $aq^k$) $= S(a) + S(ar + b)$. Similarly $S(an+b') = S(a) + S(ar + b')$. Hence we have $S(a) + S(ar + b) \equiv S(a) + S(ar + b') + c \pmod m$ and so $S(ar + b) \equiv S(ar + b') + c \pmod m$. Letting $k$ be big enough and varing $r$ from $r = 0$ to $M$, we get the desired result.

Was @above's solution the true purpose of this problem? As it is stated, both the mod $m$ condition and the $|b-b'| \ge a$ condition are irrelevant and seems very strange that such a problem would appear in a China TST.