Constacyclic BCH codes have been widely studied in the literature and have been used to construct quantum codes in latest years. However, for the class of quantum codes of length $$n=q^{2m}+1$$n=q2m+1 over $$F_{q^2}$$Fq2 with q an odd prime power, there are only the ones of distance $$\delta \le 2q^2$$?≤2q2 are obtained in the literature. In this paper, by a detailed analysis of properties of $$q^{2}$$q2-ary cyclotomic cosets, maximum designed distance $$\delta _\mathrm{{max}}$$?max of a class of Hermitian dual-containing constacyclic BCH codes with length $$n=q^{2m}+1$$n=q2m+1 are determined, this class of constacyclic codes has some characteristic analog to that of primitive BCH codes over $$F_{q^2}$$Fq2. Then we can obtain a sequence of dual-containing constacyclic codes of designed distances $$2q^2 2q^2$$d>2q2 can be constructed from these dual-containing codes via Hermitian Construction. These newly obtained quantum codes have better code rate compared with those constructed from primitive BCH codes.