Question

does a situation occurs when two particles do not exert gravitational force on each other

Answer 1

It may be occurs in the last day of earth...

Answer 2

Well, According to our current understanding ( at least upto my best knowledge), anything which has energy can create gravity.

Rᵢᵣ−1/2Rgᵢᵣ=8(pi)Tᵢᵣ

All the terms on left hand side are related to the curvature in spacetime and the term on right hand side (stress energy momentum tensor specifically) is related to the energy ( Or mass, according to mass equivalence principle). This equation simply says that anything that has mass or in other words energy, can interfere with the geometry of spacetime.

So according to general relativity, all particles are capable of creating gravity.