2-color QCD (SU(2) gauge theory coupled to fundamental fermions) has several novel features: for instance, enhanced Pauli-Gursey symmetry yields degeneracies between mesons and di/tetra-quark states. The quantum mechanical matrix model provides a simplified platform to directly probe the properties of low-energy (spin-0 and spin-1) hadrons. Using variational calculation, we numerically obtain the energy eigenstates and eigenvalues of the matrix model at ultra-strong coupling. In chiral limit, the effects of non-perturbative axial anomaly are quantified. Interestingly, in chiral limit, gluons contribute significantly to spin of hadrons. These effects are suppressed in heavy quark limit. Further, at strong coupling, the system can undergo quantum phase transitions (in presence or absence of chemical potential). The ground state can be a spin-1 di-quark state which spontaneously breaks spatial rotational symmetry.