Not quite, this isn't about acceleration vs motion, but that motion is relative in an inertial frame of reference.

If you are accelerating, then you're not in an inertial frame of reference, so it's not the same as the distant stars exceeding c.

If something else is accelerating, then that is relative, and can be viewed differently depending on the frame of reference.

As a semi-practical example, a plane taking off on an aircraft carrier. Let's have him take off backwards for the purposes of explanation, even though they wouldn't practically.
Let's say an outside observer is watching the plane take off, going back off a ship travelling forwards at 30kph. To the outside observer the plane accelerating from 30kph, through 0, and then out to -170kph.
But from someone on the ship, the plane has gone from 0 to 200kph.
That acceleration is still relative in both of those frames of reference.
But the pilot in the plane is not in an inertial frame of reference, and so it's not the same as the world accelerating under him (eg he experiences inertial G forces).