Suppose that $G:M\leftrightarrow N: U$ is a Quillen equivalence between two model categories. Suppose that $a\in M$ is a fibrant object, is it true that there exists always a fibrant object $b\in N$ and weak equivalence $a\rightarrow U(b)$ in $M$ ?


We are not assuming cofibrancy property on $a$ and $b$. 



Quillen equivalence, fibrant objects
Here is a counter-example to the dual assertion (so that you can get a counter-example to your original question by taking the opposite model categories). Consider the category ${\rm Set\_\Delta}$ of simplicial sets with the Kan-Quillen model structure. Since ${\rm Set}\_{\Delta}$ is right proper and the inclusion $i\_0:\Delta^{\{0\}} \subseteq \Delta^1$ of the $0$-vertex is a weak equivalence the adjunction
$$ (i\_0)\_!:{\rm Set}\_{\Delta} = ({\rm Set}\_{\Delta})\_{/\Delta^{\{0\}}} \rightleftarrows ({\rm Set}\_{\Delta})\_{/\Delta^1}: i^\*\_0 $$
is a Quillen equivalence, where the right hand side is endowed with the model structure induced on the slice category by the model structure of ${\rm Set}\_{\Delta}$. Here the left adjoint $(i\_0)\_!$ sends a simplicial set $X \in {\rm Set}\_{\Delta}$ to $X \to \Delta^{\{0\}} \subseteq \Delta^1$ and the right adjoint $i\_0^\*$ takes a simplicial set $Y \to \Delta^1$ over $\Delta^1$ to $Y \times\_{\Delta^1} \Delta^{\{0\}}$. Now let $Y$ be any non-empty simplicial set and consider the object of $({\rm Set}\_{\Delta})\_{/\Delta^1}$ given by $Y \to \Delta^{\{1\}} \subseteq \Delta^1$. Then $Y \to \Delta^1$ admits no maps from objects of the form $(i\_0)\_!(X)$ unless $X=\emptyset$, and hence no weak equivalences from objects of the form $(i\_0)\_!(X)$.


