  We study a superconductor Josephson junction with a Bogoliubov Fermi surface,
employing McMillan's Green's function technique. The low-energy degrees of
freedom are described by spinless fermions (bogolons), where the characteristic
feature appears as an odd-frequency pair potential. The differential equation
of the Green's function is reduced to the eigenvalue problem of the
non-Hermitian effective Hamiltonian. The physical quantities such as the
density of states and pair amplitude are then extracted from the obtained
Green's function. We find that the zero energy local density of states at the
interface decreases as the relative phase of the Josephson junction increases.
This decrease is accompanied by the generation of an even-frequency pair
amplitude near the interface. We also clarify that the $\pi$-junction-like
current phase relation is realized in terms of bogolons. In contrast to
conventional $s$-wave superconductor junctions, where even-frequency pairs
dominate in the bulk and odd-frequency pairs are generated near the interface,
our findings illuminate the distinct behaviors of junctions with Bogoliubov
Fermi surfaces. We further explore spatial dependences of these physical
quantities systematically using quasiclassical Green's functions.
