One answer to your question comes from the paper *[The Weber-Seifert dodecahedral space is non-Haken][1]* by Burton, Rubinstein, and Tillmann. 

An earlier example is (say) the $(1, 2)$-Dehn filling of the figure-eight knot.  This manifold is hyperbolic (4.22) and non-Haken (4.41).  The references are page numbers in [Thurston's lecture notes][2].  We can use [SnapPy][3] to find presentations of the fundamental groups.

Before filling:

```
In[1]: M = Manifold("4_1")
In[2]: M.fundamental_group()
Out[2]: 
Generators:
   a,b
Relators:
   abbbaBAAB
```

After filling:

```
In[3]: M.dehn_fill((1, 2))
In[4]: M.fundamental_group()
Out[4]: 
Generators:
   a,b
Relators:
   abbbaBAAB
   abAbaBabAbaBAB
```

----

**Edit**: I remembered that [Regina][4] has the Weber-Seifert manifold as one of its examples.  So, using the triangulation isomorphism signature, we can import this to SnapPy and find a presentation for its fundamental group. (We can also compute a presentation using Regina - but I prefer the notation for relations used by SnapPy).

```
In[5]: WS = Manifold("xvLvvvwMvQPPQQQQcehpjtqksntrtvoupwpsuwsvwcgacalvucahatbhapaggjgfk")
In[6]: WS.fundamental_group()
Out[6]: 
Generators:
   a,b,c,d
Relators:
   aDcbcdabdbbc
   aDabCADCbc
   abdbcdabdACBC
   aDBcdaDabbc
In[7]: WS.volume()
Out[7]: 11.1990647408
In[8]: WS.homology()
Out[8]: Z/5 + Z/5 + Z/5
```

The homology has rank three, so at least three generators are needed.  It seems to be [open][5] (?) to compute the minimal number of generators.


  [1]: https://arxiv.org/abs/0909.4625
  [2]: https://archive.org/embed/ThurstonTheGeometryAndTopologyOfThreeManifolds
  [3]: https://snappy.computop.org/
  [4]: https://regina-normal.github.io/
  [5]: https://mathoverflow.net/questions/146529/heegaard-genus-of-the-hyperbolic-dodecahedral-space-is-it-3-or-4