  We investigate the ground state of the SU($4$) Heisenberg model on a square
lattice with spatial anisotropy on each plaquette bond using the tensor-network
method based on infinite projected entangled pair states. We find that the
SU($4$) singlet ground state appears in the strongly anisotropic limit, whereas
N\'eel and valence bond crystal orders coexist in the nearly isotropic limit.
By examining the intermediate parameter region, we identify a phase transition
between these phases. The nature of the phase transition is likely to be of
first order, and the transition point is estimated to be around $J'/J\approx
0.85(5)$, where $J$ and $J'$ are the interaction strengths of intra- and
inter-plaquette bonds, respectively. We also calculate the anisotropy
dependence of singlet correlations on a plaquette bond, which will be useful
for future experiments of ultracold atoms in optical lattices.
