Bound states in the continuum (BICs) enable unique features in tailoring light-matter interaction on nanoscale. These radiationless localized states drive theoretically infinite quality factors and lifetimes for modern nanophotonics, making room for a variety of emerging applications. Here we use the peculiar properties possessed by the so-called PT-symmetric optical structures to propose the novel mechanism for the quasi-BIC manifestation governed by the PT-symmetry breaking. In particular, we study regularities of the spontaneous PT-symmetry breaking in trilayer structures with the outer loss and gain layers consisting of materials with permittivity close to zero. We reveal singular points on the curves separating PT-symmetric and broken-PT-symmetry states in the parametric space of the light frequency and the angle of incidence. These singularities remarkably coincide with the BIC positions at the frequency of volume plasmon excitation, where the dielectric permittivity vanishes. The loss and gain value acts as an asymmetry parameter that disturbs conditions of the ideal BIC inducing the quasi-BIC. Fascinating properties of these quasi-BICs having ultrahigh quality factors and almost perfect transmission can be utilized in sensing, nonlinear optics, and other applications.