  In this work, we introduce and construct specific $q$-polynomials that are
desired from the well-established families of $q$-orthogonal polynomials,
namely little $q$-Jacobi polynomials and $q$-Laguerre polynomials,
respectively. We examine these newly constructed $q$-polynomials and observe
that they possess integral representations of little $q$-Jacobi polynomials and
$q$-Laguerre polynomials. These polynomials solve a third-order $q$-difference
equation and display an unconventional four-term recurrence relation. This
unique recurrence relation makes us categorize them as $q$-Sobolev-type
orthogonal polynomials. This motivation leads to defining the general
Sobolev-type orthogonality for $q$-polynomials. Special cases of these
polynomials are also explored and discussed. Furthermore, we delve into the
behavior of these $q$-orthogonal polynomials of Sobolev type as the parameters
approach $1$. We also examine their zeros and interlacing properties.
