Answer
No.
Work Step by Step
We know that the moment of inertial of an object is given by
$$I=\int r^2dm$$
where $r$ is the object's dimension around some point we need to measure the object's moment of inertia at, and $dm$ is the distribution of the mass.
For simplicity, take the moment of inertia of a disk.
$$I_{disk}=\frac{1}{2}mR^2$$
where $m$ is the disk's mass and $R$ is its radius.
This formula shows that the moment of inertia has no relation to the angular speed at all. It has no relation to whether the object is rotating or not.
So, no, the object has a moment of inertia in both cases. It is representing some physical properties of the object.