The momentum of an object is the product of the mass of the object and the velocity at which it is moving. To measure the total momentum of the object-Earth system, the reference point should be taken as one lying outside the system.
As the object moves towards the Earth, it is accelerated and its velocity towards the Earth increases. At the same time, from the external reference point is can be seen that the Earth accelerates towards the object and there is an increase in its velocity towards the object. The magnitude of the velocity of the Earth is very small as the mass of the Earth is very large.
The total momentum of the object-Earth system is the sum of the momentum of the object and the Earth. As the velocity of both is in the opposite direction, there is no net change in the momentum. The total momentum of the object-Earth system remains the same as the object is dropped towards the Earth