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 $\mathbb Q$.

8$\begingroup$ What do you mean by "explicit"? You can take presentation complex for $Q$, immerse it in $R^4$ and then pullback regular neighborhood. The result is your manifold. $\endgroup$– MishaJul 1 '13 at 13:37

$\begingroup$ See [this stackexchange question and answers.][1] [1]: math.stackexchange.com/questions/36775/… $\endgroup$– Igor RivinJul 1 '13 at 17:12
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 suffices to take $X=M \times \mathbb{R}$.