The interest in hyperbolic metamaterials is fueled by fascinating optical properties exhibited by this class of artificial media. Their optical features originate from hyperbolic dispersion emerging due to the shape anisotropy of the metal-dielectric composite. In this work, we study experimentally and numerically the second-harmonic generation (SHG) in ordered arrays of Au nanorods embedded in porous aluminum oxide. Strong increase of the SHG intensity in the vicinity of the epsilon-near-zero (ENZ) spectral point accompanied by dramatic phase modulation of the SHG wave is revealed. These effects are attributed to resonant enhancement of the electric field of the light wave and transition from the elliptical to hyperbolic dispersion law in hyperbolic metamaterials near the ENZ point.