In this manuscript, generating functions are constructed for the new special families of polynomials and numbers using the p-adic q-integral technique. Partial derivative equations, functional equations and other properties of these generating functions are given. With the help of these equations, many interesting and useful identities, relations, and formulas are derived. We also give p-adic q-integral representations of these numbers and polynomials. The results we have obtained for these special numbers and polynomials are closely related to well-known families of polynomials and numbers including the Bernoulli numbers, the Apostol-type Bernoulli numbers and polynomials and the Frobenius-Euler numbers, the Stirling numbers, and the Daehee numbers. We give some remarks and observations on the results of this paper.