  We show that spontaneous time-reversal-symmetry (TRS) breaking can naturally
arise from the interplay between pair density wave (PDW) ordering at multiple
momenta and nesting of Fermi surfaces (FS). Concretely, we consider
pair-density-wave superconductivity on a hexagonal lattice with nested FS at
$3/4$ electron filling, which is related to a recently discovered
superconductor CsV$_3$Sb$_5$. Because of nesting of the FS, each momentum $\bk$
on the FS has at least two counterparts $-\bk\pm\mathbf{Q}_{\alpha}$
($\alpha=1,2,3$) on the FS to form finite momentum ($\pm\mathbf{Q}_{\alpha}$)
Cooper pairs, resulting in a TRS and inversion broken PDW state with stable
Bogoliubov Fermi pockets. Various spectra, including (local) density of states,
electron spectral function and the effect of quasi-particle interference, have
been investigated. The partial melting of the PDW will give rise to
$4\times{}4$ and $\frac{4}{\sqrt{3}}\times\frac{4}{\sqrt{3}}$ CDW orders, in
addition to the $2\times2$ CDW. Possible implications to real materials such as
CsV$_3$Sb$_5$ and future experiments have been further discussed.
