Let \(q\ge 3\) be an odd prime power and \(m\ge 2\) be an even integer. In this paper, a kind of narrow-sense constacyclic BCH codes over \(\mathbb {F}_{q^2}\) with lengths \(n=\frac{q^{2m}-1}{2a(q+1)}\) are studied, where \(q\equiv a+1 \pmod {2a}\) and \(a<q-1\) . The maximum designed distances such that narrow-sense constacyclic BCH codes over \(\mathbb {F}_{q^2}\) with length \(\frac{q^{2m}-1}{2a(q+1)}\) containing their Hermitian dual codes are determined. We obtain some quantum codes with good parameters by such narrow-sense constacyclic BCH codes. The quantum codes that we construct have larger designed distances or dimensions than the ones with the same length in the literature.
