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. 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.
