Could an object have no gravitational potential energy on earth since its impossible not to have any gravity

GPE = MGH. (mass x gravity x height).
If this object is on the ground, it still has GPE because gravity is acting on it and the ground is preventing it falling any further. That GPE is not likely to be turned into kinetic energy, unless it is allowed to fall into a hole or something.
I think if I untangle your question, an object on earth would have NO GPH if it fell into a very deep (theoretical) hole and reached the centre of mass of the world. All the GPE would be changed into kinetic energy, which would dissipate as the object neared the CM of earth.
At the CM, there is no net gravity, as gravitational forces are the same in every direction. Remove gravity and/ or height, or both, you have no GPE.

Yes you are right every object having mass has negative potential energy with respect to object if they had been very far off or at infinity, where potential energy has been taken as zero.