The value of Q relates the impedance of the inductor to its resistance:
Q = X(L)/R, so for a frequency f,
Q = 2 pi f L / R.
For a series connection both L and R sum, so
Q' = 2 pi f (2L) / 2R
The factors of 2 cancel, and the Q of the composite inductor (Q') is identical to the Q of the component inductors (for the case identical Qs).
The same calculation can be made for any number of inductors, in series or in parallel. In all cases, the composite Q will lie somewhere between the highest and lowest Qs of the individual components.