The lattice constants, energy band properties and phase stabilities of wurtzite (WZ) and rocksalt (RS) Be1-xMgxO alloys are investigated by the plan-wave pseudopotential method in the generalized gradient approximation based on the density functional theory. The theoretical results show that the lattice constants of WZ and RS Be1-xMgxO alloys increase and their bandgaps decrease as the content x of Mg increases from 0 to 1. For the same Mg content values, the bandgap of RS Be1-xMgxO alloy is greater than that of WZ Be1-xMgxO alloy. The WZ phase will transit to the RS phase when the Mg content is about 0.89. In order to obtain the theoretical values in accordance with the experimental results, the bandgaps of WZ and RS Be1-xMgxO alloys are corrected and the values of bandgap bowing parameter b are 3.451 eV and 4.96 eV for WZ Be1-xMgxO and RS Be1-xMgxO respectively. The reason of large band gap bowing parameter b is attributed to a large difference in ionic radius between Be and Mg. Besides, the relations among energy bandgap, bowing parameter and lattice constant of wurtzite BeO-ZnO-CdO ternary alloy are analyzed.