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In this article, we investigate the existence of the positive solutions tothe following class of quasilinear {Schr\"odinger} equations involvingStein-Weiss type convolution  \begin{align*}  -\Delta_N u -\Delta_N (u^{2})u +V(x)|u|^{N-2}u=  \left(\int_{\mathbbR^N}\frac{F(y,u)}{|y|^\beta|x-y|^{\mu}}~dy\right)\frac{f(x,u)}{|x|^\beta} \;\;\text{ in}\;  \mathbb R^N,  \end{align*} where $N\geq 2,\,$ $0<\mu<N,\, \beta\geq 0,$ and $2\beta+\mu\leqN.$ The potential $V:\mathbb R^N\to \mathbb R$ is a continuous functionsatisfying $0<V_0\leq V(x)$ for all $x\in \mathbb R^N$ and some appropriateassumptions. The nonlinearity $f:\mathbb R^N\times \mathbb R\to \mathbb R$ is acontinuous function with critical exponential growth in the sense of theTrudinger-Moser inequality and $F(x,s)=\int_{0}^s f(x,t)dt$ is the primitive of$f$.

Quasilinear Schr\"odinger equations with Stein-Weiss type convolution and critical exponential nonlinearity in $\mathbb R^N$