What is the origin of using Q to denote a set of states in an automaton?
The symbol $q$ to denote the state of a finite automaton appears (for the first time??) in Moore's 1956 paper Gedanken-Experiments on Sequential Machines and since the paper became very popular, I suppose that Moore's notation has been adopted since then. Apparently, Moore didn't use any special notation for the set of states, but it you adopt $q$ for a state, $Q$ seems to be an obvious choice for the set of states. 

I don't think there is any special meaning for the letter $q$ (or $Q$).

[1] E.F. Moore, Gedanken experiments on sequential machines. In C. E. Shannon and J. McCarthy, editors, Automata Studies, 129–156. Princeton University Press, 1956.