This might be well known for algebraic topologist. So I am looking for an explicit example of a 4 dimensional manifold with fundamental group isomorphic to the rationals $\mathbb Q$.

Since any compact manifold has the homotopy type of a finite CWcomplex (see this MathOverflow question: Are nonPL manifolds CWcomplexes?) and $\mathbb{Q}$ is not finitely presented, the manifold $X$ you are looking for is necessarily noncompact. An explicit construction of a noncompact threemanifold $M$ with $\pi_1(M)=\mathbb{Q}$ can be found in the paper B. Evans and L. Moser: Solvable Fundamental Groups of Compact 3Manifolds, Transactions of the American Mathematical Society 168 (1972), see in particular page 209. Now it sufficies to take $X=M \times \mathbb{R}$. 

