Potential energy surfaces of 6 electronic states of TlH2 and InH2 and 8 electronic states of TlH2+ and InH2+ are computed. In addition the ground states of TlH3, InH3, TlH3+, InH3+, TlH, and TlH+ are investigated. A complete active space multiconfiguration self-consistent field (CAS-MCSCF) followed by second-order configuration interaction (SOCI) and relativistic configuration interaction (RCI) including spin-orbit coupling calculations are carried out. The step-wise bond energies, D(e)(H(n-1)M-H) and adiabatic ionization potentials are computed. The ground states of TlH2 and InH2 are found to be bent (2A1; theta-e approximately 121.5-degrees, 120-degrees) while the ground states of TlH2+ and InH2+ are linear (1-SIGMA-g+). The ground states of TlH3 and InH3 are found to be 1A1 (D3h) states while the ground states of TlH3+ and InH3+ are Jahn-Teller distorted B2(2)(C2-upsilon) states. The unique bond length of TlH3+ and InH3+ is shorter than the two equal bond lengths. The bond angles (H-M-H) for TlH3+ and InH3+ deviate considerably from the neutral theta-e = 120-degrees to near 69-degrees. The TlH+ ion is found to be only 0.04 eV stable. Periodic trends in the geometries, bond energies and IPs are studied. Spin-orbit effects were found to be significant for TlH(n) species. The IPs of InH(n) and TlH(n) exhibit odd-even alternation. The bond energies also show an interesting trend as a function of n.