Can an acid react with its conjugate base ?

No, an acid cannot react with its conjugate base.

The chemical equation corresponding to an acid and its conjugate base is :
Acid ⇌ H⁺ + (Conjugate base)

The acid and its conjugate base are on the two sides of the chemical equation.
If the acid reacts with its conjugate base, the above equation cannot be established.

Of course it can. But there is no noticeable change. You'll never know any protons were exchanged.