$S(m,q)$ is a hypergeometric function (you have take the upper limit of the sum $\infty$, as terms for $i$ bigger than $m-q$ will all vanish); a "standard" method would be to find it explicitly, and then to use a representation of it by an integral, which can be estimated by methods from asymptotic analysis.
EDIT: I am referring to the standard technique to identify a hypergeometric series explained in e.g. Chapter 3 of the book "A=B" by Petkovsek, Wilf, and Zeilberger.
Using it you will be able to write $$ S(m,q)=\binom{m}{q}^2 \ _3F_0(-m,q-m,q-m;-;\frac{-1}{(q+1)^2}). $$