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.
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