The momentum of individual objects in a collision does change (no matter whether it's elastic or inelastic). However, the total momentum is conserved (does not change), again, irrespective of the fact that the collision is elastic or inelastic. Thus there is a non zero, and in fact equal and opposite impulse on both the objects. Due to this equal and opposite impulse (caused by the normal force between them), the net impulse on the system of those two objects becomes zero and thus the net momentum of the system is conserved.

The momenta of individual objects in a collision do change (no matter whether it's elastic or inelastic). However, the total momentum is conserved (does not change), again, irrespective of the fact that the collision is elastic or inelastic. Thus there is a non zero, and in fact equal and opposite impulse on both the objects. Due to this equal and opposite impulse (caused by the normal force between them), the net impulse on the system of those two objects becomes zero and thus the net momentum of the system is conserved.