Accelerating and not accelerating are concepts tied to your reference frame. This is a mathematical construction which a physicist constructs with which to solve problems. Your acceleration can change with respect to these systems. To explore this, we should start with a simple case and add complexity until we come full circle to your single object universe.
In a world with multiple objects, it is easy to see how this works. If I have a bowling ball falling to the ground, I can make a reference frame such that the ground is holding still, and see that the ball is accelerating in that frame, and the ground is not. I can also make a reference frame from the point of view of the ball. In this frame, it is the ground that is accelerating towards the ball, and the ball is motionless.
In general, we find these to be the most convenient reference frames. However, sometimes more exotic frames are useful. For example, in the above example with the ground reference frame, we ignore the effect of the ball pulling on the Earth. A bowling ball doesn't have much pull. It will pull the earth nanometers at most as they come together. But the earth does indeed move. When studying astonomical objets, this matters. Consider our solar system. We like to say that Jupiter revolves around the sun, so it would make sense to create a coordinate system where the sun is fixed and Jupiter is in motion, right? Well, almost. If we did this, we'd see Jupiter "wobbling" for some not-fully explained reason.
The reason for this wobble is that Jupiter does not precisely orbit the sun. Objects actually revolve around the center of mass of the whole system, known as the "barycenter." In our case, Jupiter is actually massive enough that the barycenter isn't at the center of the sun. Sometimes it's actually outside of the photosphere of the sun (though it never actually gets far enough to leave the corona). If we create a new reference frame centered on this barycenter, we find the orbits are much simpler. Both Jupiter and the sun orbit around the barycenter of the solar system in nice clean orbits (ignoring the very minor effect of all non-Jupiter planets).
But note that I just built a reference frame around the barycenter of the solar system. The barycenter is not an object. There's no diamond crystal at that point marking the barycenter. It's just a mathematical concept. I built a reference frame around a mathematical construct, rather than a physical one. You're allowed to do that.
So going back to your one object system, I can build any reference frame I please. I can make a reference frame which would follow the trajectory of some hypothetical rocket as it streaks off into space, whether or not that rocket actually exists. The math which we use to predict the motion of things in our universe will work no matter what reference frame you pick. And in that reference frame, our one object in the universe will indeed be accelerating.
Now such exotic reference frames would typically not be considered practical. I would expect any reasonable practical problem would end up constructing a reference frame centered on the one object in the universe, because that creates a symmetric situation and it's much easier to do math when there is nice clean symmetries. Anyone using a difference reference frame will probably be asked to justify their choice, but the math will still work.