An object at rest does not have an acceleration does it?
At rest in what frame of reference? A universal inertial frame of reference does not exist.
What accelerometers measure is the acceleration due to the sum of all of the accelerations due to the real non-gravitational forces acting on an accelerometer. Accelerometers do not sense acceleration due to fictitious forces such as centrifugal acceleration and the Coriolis effect due to the Earth's rotation. Accelerometers do not sense acceleration due to gravity; no device can per Einstein's equivalence principle.
Consider an accelerometer at rest on the surface of the Earth. The forces acting on it from the perspective of an Earth-fixed frame of reference are gravity, centrifugal force, buoyancy by the air, and the normal force exerted by the Earth. The accelerometer does not measure gravitation or centrifugal acceleration. All that are left is the tiny buoyancy force and the much larger normal force. An accelerometer at rest on the surface of the Earth registers an acceleration of about one g, upward.
Next consider an accelerometer orbiting the Earth at an altitude of 6378 km. There's a tiny, tiny bit of air even that high, but it is so very sparse that atmospheric drag is essentially non-existent. Even the very best accelerometer will not be able to sense that tiny atmospheric drag. Solar radiation pressure also acts on the accelerometer, but this too is very small. The only other forces acting on this accelerometer are the gravitational force exerted by the Earth, the Sun, the Moon, other planets, etc. Accelerometers do not measure gravity. This accelerometer registers zero, or very close to zero.