No, it is not Haken. 

You can test this yourself by looking for incompressible normal surfaces with respect to a triangulation of the manifold, this functionality is included in [Regina][1]. Moreover, the Weeks manifold is a premade example so determining if it is Haken can be done directly from within the Recognition tab. This completes in a few seconds on my laptop. You can read more about the normal surface theory algorithms used by Regina to do this in the [documentation][2].

![Testing within Regina 4.96][3]


  [1]: http://regina.sourceforge.net/
  [2]: http://regina.sourceforge.net/docs/tri-analysis.html#tri-recognition
  [3]: https://i.sstatic.net/XGWz3.png