By the principle of differential subordination and the $q$-derivative operator, we introduce the $q$-analog ${\mathcal{S}\mathcal{P}}_{snail}^q(\lambda ;\alpha ,\beta ,\gamma )$ of certain class of analytic functions associated with the generalized Pascal snail. Firstly, we obtain the coefficient estimates and Fekete-Szegö functional inequalities for this class. Meanwhile, we also estimate the corresponding symmetric Toeplitz determinant. Secondly, for all the above results we provide the corresponding results for the reduced classes ${\mathcal{S}\mathcal{P}}_{snail}^q(\alpha ,\beta ,\gamma )$ and ${\mathcal{R}\mathcal{P}}_{snail}^q(\alpha ,\beta ,\gamma )$. Thirdly, we characterize the Bohr radius problems for the function class ${\mathcal{S}\mathcal{P}}_{snail}^q(\alpha ,\beta ,\gamma )$. Lastly, we establish certain results for some new subclasses of functions defined by the neutrosophic Poisson distribution series.