Yes, it would have velocity, or at least all the mechanical properties of it. That velocity would be undefined from our perspective since all possible ways you could define it (0, 5 m/s to the right, etc.) would be equally valid depending on the coordinates you choose to put at rest, but that’s a measurement problem, not anything intrinsic to reality.

The important bits are: 1) inertia exists as long as you have the minimum ingredients, an object in spacetime, 2) all possible inertial reference frames are still meaningfully distinct from each other and stay as such, 3) any change in velocity of the object would show up in all possible inertial reference frames, and 4) any such change will stay logically consistent regardless of which inertial reference frame you choose to measure the velocity change in.