In this paper we focus on continuous univariate probability distributions, like McKay distributions, $K$-distribution, generalized inverse Gaussian distribution and generalised McKay distributions, with support $[0,\infty),$ which are related to modified Bessel functions of the first and second kinds and in most cases we show that they belong to the class infinitely divisible distributions, self-decomposable distributions, generalized gamma convolutions and hyperbolically completely monotone densities. Some of the results are known, however the proofs are new and we use special functions technique. Integral representations of quotients of Tricomi hypergeometric functions as well as of quotients of Gaussian hypergeometric functions, or modified Bessel functions of the second kind play an important role in our study. In addition, by using a different approach we rediscover a Stieltjes transform representation due to Hermann Hankel for the product of modified Bessel functions of the first and second kinds and we also deduce a series of new Stieltjes transform representations for products, quotients and their reciprocals concerning modified Bessel functions of the first and second kinds. By using these results we obtain new infinitely divisible modified Bessel distributions with Laplace transforms related to modified Bessel functions of the first and second kind. Moreover, we show that the new Stieltjes transform representations have some interesting applications and we list some open problems which may be of interest for further research. In addition, we present a new proof via the Pick function characterization theorem for the infinite divisibility of the ratio of two gamma random variables and we present some new Stieltjes transform representations of quotients of Tricomi hypergeometric functions.