We consider the Hecke groups generated by $ S(z)=z+\lambda $ and $ T(z)=-1/z $ with $ \lambda=2\cos(\pi/q) $ for $ q\geqq 3 $ . We show that when $ q=4 $ or 6, the units in $ \mathbb{Z}[\lambda] $ are infinite pure periodic $ \lambda $ -fractions, and hence cannot be cusp points (images of $ \infty $ by a member of the group.) The case when $ q=7 $ is quite different; examples of units that are finite $ \lambda $ -fractions and units that are infinite $ \lambda $ -fractions are given. We conclude with a conjecture on the structure of these infinite repeating units.