We investigate the contribution of the beamed jet component to the high-energy emission in young and compact extragalactic radio sources, focusing for the first time on the -ray band. We derive predictions on the -ray luminosities associated with the relativistic jet assuming a leptonic radiative model. The high-energy emission is produced via Compton scattering by the relativistic electrons in a spherical region at the considered scales (${\sim }^{<}$10 kpc). Simulations show a wide range of -ray luminosities, with intensities up to$ ~10^{46}-10^{48}erg s^{-1}$ depending on the assumed jet parameters. We find a highly linear relation between the simulated X-ray and -ray luminosities that can be used to select candidates for -ray detection. We compare the simulated luminosity distributions in the radio, X-ray, and -ray regimes with observations for the largest sample of X-ray-detected young radio quasars. Our analysis of ~4-yr Fermi Large Area Telescope (LAT) data does not yield any statistically significant detections. However, the majority of the model-predicted -ray fluxes for the sample are near or below the current Fermi -LAT flux threshold and compatible with the derived upper limits. Our study gives constraints on the minimum jet power ($ L{jet, kin}/L_{disc} > 0.01_{\sim }^{<}1 kpc$) and on the particle-to-magnetic field energy density ratio that are in broad agreement with equipartition assumptions.