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

An earlier and "easier" example is 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][1].  

Finally, we can use SnapPy 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
```

[1]: https://archive.org/embed/ThurstonTheGeometryAndTopologyOfThreeManifolds