Different regimes are usually observed for fluid migration through an immersed granular layer. In this work, we report a puzzling behavior when injecting water at a constant flow rate through a nozzle at the bottom of an immersed granular layer in a Hele-Shaw cell. In a given range of parameters (granular layer height and fluid flow rate) the granular bed is not only fluidized, but the particle-laden jet also exhibits periodic oscillations. The frequency and amplitude of the oscillations are quantified. The Strouhal number displays a power-law behavior as a function of a nondimensional parameter, J, defined as the ratio between the jet velocity at the initial granular bed height and the inertial particle velocity. Fluid-particle coupling is responsible for the jet oscillations. This mechanism could be at the origin of the cyclic behavior of pockmarks and mud volcanoes in sedimentary basins.