The quantum antiferromagnet Cu2Te2O5Br2 was investigated by NMR and nuclear quadrupole resonance (NQR). The Te-125 NMR investigation showed that there is a magnetic transition around 10.5 K at 9 T, in agreement with previous studies. From the divergence of the spin-lattice relaxation rate, we ruled out the possibility that the transition could be governed by a one-dimensional divergence of the spin-spin correlation function. The observed anisotropy of the Te-125 shift was shown to be due to a spin polarization of the 5s(2) "E" doublet of the [TeO3E] tetrahedra, highlighting the importance of tellurium in the exchange paths. In the paramagnetic state, Br NQR and NMR measurements led to the determination of the Br hyperfine coupling and the electric field gradient tensor, and to the spin polarization of Br p orbitals. The results demonstrate the crucial role of bromine in the interaction paths between Cu spins.