In this paper, a special class of two-generator quasi-twisted (QT) codes with index 2 will be presented. We explore the algebraic structure of the class of QT codes and the form of their Hermitian dual codes. A sufficient condition for self-orthogonality with Hermitian inner product is derived. Using the class of Hermitian self-orthogonal QT codes, we construct two new binary quantum codes [[70, 42, 7]]₂, [[78, 30, 10]]₂. According to Theorem 6 of Ref.[2], we further can get 9 new binary quantum codes. So a total of 11 new binary quantum codes are obtained and there are 10 quantum codes that can break the quantum Gilbert-Varshamov (GV) bound.