  The Neptunian satellite system is unusual, comprising Triton, a large
($\sim2700$ km) moon on a close-in, circular, yet retrograde orbit, flanked by
Nereid, the largest irregular satellite ($\sim$300 km) on a highly eccentric
orbit. Capture origins have been previously suggested for both moons. Here we
explore an alternative in-situ formation model where the two satellites
accreted in the circum-Neptunian disk and are imparted irregular and eccentric
orbits by a deep planetary encounter with an ice giant (IG), like that
predicted in the Nice scenario of early solar system development. We use
$N$-body simulations of an IG approaching Neptune to 20 Neptunian radi
($R_\mathrm{Nep}$), through a belt of circular prograde regular satellites at
10-30 $R_\mathrm{Nep}$. We find that half of these primordial satellites remain
bound to Neptune and that 0.4-3\% are scattered directly onto wide and
eccentric orbits resembling that of Nereid. With better matches to the observed
orbit, our model has a success rate comparable to or higher than capture of
large Nereid-sized irregular satellites from heliocentric orbit. At the same
time, the IG encounter injects a large primordial moon onto a retrograde orbit
with specific angular momentum similar to Triton's in 0.3-3\% of our runs.
While less efficient than capture scenarios (Agnor & Hamilton 2006), our model
does indicate that an in-situ origin for Triton is dynamically possible. We
also simulate the post-encounter collisional and tidal orbital evolution of
Triton analogue satellites and find they are decoupled from Nereid on
timescales of $\sim$$10^4$ years, in agreement with Cuk & Gladman (2005).
