Measurements of the cross sections are conducted for coherent photoproduction of the neutral pion and eta meson on the d, gamma deutron to pi0 eta deutron, at the incident energy in the range of the reaction threshold to 1.15 GeV. A rapidly increasing trend below 1 GeV is observed in the total cross section. The data are effectively reproduced by theoretical calculations incorporating the meson-deutron final-state interactions. However, the measured deutron angular distribution d sigma/d Omegad in the center-of-mass frame (gamma d) is rather flat, and it differs significantly from the strong backward-peaking behavior expected in the kinematics of d formation after pi0 eta photoproduction on an initial bound nucleon. In addition, the d sigma/dMeta d differential cross section exhibits a prominent enhancement near the eta d threshold, while d sigma/dMpi d exhibits an enhancement near the known pi d resonance with I=1 and JP=2+. From these observations, we have concluded that two reaction sequences occur; gamma d to pi0 D01 to pi0 eta d and eta D12 to pi0 eta d, where DIJ denotes a state with a baryon number of 2, an isospin of I, and a spin of J. The enhancement corresponding to D01 is expected to be a theoretically predicted eta NN bound state with I=0 and JP=1- or a virtual state of eta d. It is found that attraction between eta d is certainly very strong.