Can particles gain energy during inelastic collision?

We know that when two particles collides inelastically, the net energy equation is given by $$E_i=E_f+Q$$ where $E_i$ and $E_f$ are initial and final mechanical energies of the particles and $Q$ is heat produces.

To be technically correct, the loss in macroscopic kinetic energy equals a gain in the microscopic internal kinetic energy of the object(s), i.e., the energy of motion of the atoms and molecules of the inelastic object(s). That increase in internal kinetic energy raises the temperature of the object(s). Then a transfer of heat is possible out of the object(s) to a lower temperature environment.

In most of the cases we have $Q>0$ and thus $E_i>E_f$.

That is correct, unless there happens to be stored mechanical or chemical potential energy in the object that is released and converted to macroscopic kinetic energy due to the collision.

I just want to know if there are cases where $E_i>E_f$ so that there is a gain in net mechanical energy of the particles. If it is possible what are the examples?

An example of a conversion of chemical potential energy is the detonation of an explosive charge upon impact. The total kinetic energy of the exploded particles will exceed the total kinetic energy of the object (s) before the collision due to the conversion of chemical potential energy to kinetic energy. (Though momentum will still be conserved if the system is isolated).

A theoretical example (I'm not award of an actual example) is the release (conversion) of mechanical potential energy of a coil that is kept compressed by some sort of hair trigger device. A slow moving object collides with the spring and the spring decompresses propelling the object with a greater velocity than before the collision. (The spring recoils in the opposite direction and momentum is conserved).

Hope this helps.