Let $\mathcal{C}$ and $\mathcal{D}$ be two categories and $F : \mathcal{C} \to \mathcal{D}$ and $G: \mathcal{D} \to \mathcal{C}$ be two functors such that $F$ is left adjoint to $G.$ Also assume that $G$ is an exact functor and the unit map $G \circ F$ is an isomorphism. Since $F$ is a left adjoint, therefore, one may easily observe that $F$ is a right exact functor.
My question is: on the basis of the above assumption can one concludes the functor $F$ is also exact.
No we can not in that generality. The dual situation is more common: a fully faithful functor $i:\mathcal E\to \mathcal D$ is right adjoint to a left exact functor $a:\mathcal D\to \mathcal E$. (If $\mathcal D$ is a presheaf category, it is more or less a characterization for $\mathcal E$ to be a Grothendieck topos.)
The question you are asking then is: does $i$ preserve finite colimits? A priori no: suppose $\mathcal D$ is the category of presheaves over the open sets of a topological space $X$ and $\mathcal E$ is the category of sheaves on $X$; then a morphism in $\mathcal E$ is an epi as soon as it is surjective at each stalk, while its image by $i$ is an epi when it is surjective on every open set. I let you cook an explicit example based on that. (Or go fish there.)