Exact sampling methods have been recently developed for generating random variates for exponentially tilted $\alpha$-stable distributions. In this paper we show how to generate, exactly, random variates for a more general class of tilted $\alpha$-stable distributions, which is referred to as the class of Laguerre-type exponentially tilted $\alpha$-stable distributions. Beside the exponentially tilted $\alpha$-stable distribution, such a class includes also the Erlang tilted $\alpha$-stable distribution. This is a special case of the so-called gamma tilted $\alpha$-stable distribution, for which an efficient exact random variate generator is currently not available in the literature. Our result fills this gap.