The aim of this paper is to study the notion of a quarter-symmetric metric connection on an almost contact B−metric manifold (M, ϕ, ξ, η,). We obtain the relation between the Levi-Civita connection and the quarter-symmetric metric connection on (M, ϕ, ξ, η,). We investigate the curvature tensor, Ricci tensor and scalar curvature tensor with respect to the quarter-symmetric metric connection. In case the manifold (M, ϕ, ξ, η,) is a Sasaki-like almost contact B−metric manifold, we get some formulas. Finally, we give some examples of a quarter-symmetric metric connection