In his 1936 paper "ON COMPUTABLE NUMBERS, WITH AN APPLICATION TO THE ENTSCHEIDUNGSPROBLEM", Alan Turing wrote :
"We may compare a man in the process of computing a real number to
machine which is only capable of a finite number of conditions q1, q2,
.... qR which will be called " m-configurations"
So he stressed the fact that the machine has a finite, discrete (not continuous) number of states or quantities. For me, it is a reference to the term Quanta used in physics to denote phenomena variating not continuously but by "leaps" or "quanta". In his 1950 article "Computing Machinery and Intelligence" Alan Turing is more explicit about "leaps" speaking of "sudden jumps":
"The digital computers considered in the last section may be
classified amongst the "discrete-state machines." These are the
machines which move by sudden jumps or clicks from one quite definite
state to another."
So I think that Alan Turing used q instead of s to denote a machine state to stress the fact that the state machine can be only in a set of discrete and finite values like the quanta in physics. And since then, the same notation is generally used.