We show results for the Shannon entropy, Rényi entropy, Fisher information, and their relative relations in the position space for the Li1+ and Be2+ ions embedded in a strong plasma. This medium is simulated by the ion-sphere screening model as a function of the plasma electron density, N e. We solve the Hartree-Fock equations using a numerical grid method in a self-consisting manner and the simple Hylleraas-type correlated wave function is applied. Our findings show a reduction of the total energy for both ions as the electron density is increased reaching the ionization limit. We find interesting features in the description of the Shannon entropy and Fisher information where the correlation effects of the embedded ions are relevant. The Shannon-Fisher product results indicate that the electron-electron correlation decreases as N e increases and similar behavior is found for the Rényi-Fisher product. Finally, in the absence of the plasma medium, our results are in excellent agreement with those available in the literature.