Purely from an informational point of view (ie, looking at how does the
drive know what the velocity zero is, while ignoring how it interacts
with it), as far as I can see there are at most two possibilities.
The first and in my opinion by far the most likely (but only because the
other is even less likely!) possibility is zero velocity relative to the
big bang; which is also a zero relative to the mass in and/or of the
universe; and for practical purposes is very close indeed to the rest
frame relative to the cosmic microwave background.
The last is something which we can actually measure; we are travelling
at 369±0.9 km/s in the direction of galactic longitude l = 263.99±0.14°,
b = 48.26±0.03 relative to that rest frame.
The second possibility comes from General Relativity and is sort of
similar in terms of being a summation of the effects of all the mass in
the universe, but it takes local matter more into account. For various
reasons I think it's very unlikely indeed but I thought I'd mention it
as, like the rest of these speculations, it is not impossible, assuming
the rest of physics is correct but incomplete.