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.

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 or in the original paper by Jaco and Oertel.