This letter discusses the construction of 16-quadratic-amplitude modulation (QAM) Golay complementary sequences of length $N=2^{m}$. Based on the standard binary Golay–Davis–Jedwab (GDJ) complementary sequences (CSs), we present a method to convert the aforementioned GDJ CSs into the required sequences. The resultant sequences have the upper bounds $3.6N$, $2.8N$, $2N$, $1.2N$, and $0.4N$ of peak envelope powers, respectively, depending on the choices of their offsets. The numbers of the proposed sequences, corresponding to five upper bounds referred to above, are $(24m-16)(m!/2)2^{m+1}$, $128(m-1)(m!/2)2^{m+1}$, $(176m-160)(m!/2)2^{m+1}$, $128(m-1)(m!/2)2^{m+1}$, and $(24m-16)(m!/2)2^{m+1}$. Our sequences can be potentially applied to the QAM systems whose input signals are binary signals.