  The q-rational numbers and the q-irrational numbers are introduced by S.
Morier-Genoud and V. Ovsienko. In this paper, we focus on q-real quadratic
irrational numbers, especially q-metallic numbers and q-rational sequences
which converge to q-metallic numbers, and consider the radiuses of convergence
of them when we assume that q is a complex number. We construct two sequences
given by recurrence formula as a generalization of the q-deformation of
Fibonacci numbers and Pell numbers which are introduced by S. Morier-Genoud and
V. Ovsienko. We give an estimation of radiuses of convergence of them, and we
solve on conjecture of the lower bound expected which is introduced by L.
Leclere, S. Morier-Genoud, V. Ovsienko, and A. Veselov for the metallic numbers
and its convergence rational sequence. In addition, we obtain a relationship
between the radius of convergence of the [n,n,...,n,...]_q and [n,n,...,n]_q in
the case of n=3 and n=4.
