We present the first calculation of the connected scalar matrix element and the momentum fraction of the charm quark within the ${3}/{2}^{+}$ and ${3}/{2}^{-}$ triply charmed baryons on lattice QCD. The results are based on overlap valence fermions on two ensembles of $ N_f=2+1 $ domain wall fermion configurations with two lattice spacings. The corresponding sea quark pion masses are 300 MeV and 278 MeV. The separated contributions to the triply charmed baryon mass are derived through the decomposition of the QCD energy-momentum tensor. The contribution of the connected charm quark matrix element to the triply charmed baryon is about 3/2 times that of the charmonium, and it is almost 70% of the total mass. The mass splitting of ${3}/{2}^{+}$ and ${3}/{2}^{-}$ triply charmed baryons is mainly from the $ \langle H_{E}\rangle $ of the QCD energy-momentum tensor. A mass decomposition based on the quark model is also studied for comparison.