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It follows from the definitions that $[X/G]$ is representable by a scheme $S$ if and only if $X$ is a $G$-torsor over $S$, i.e. the natural map $G\times X\rightarrow X\times _SX$ given by $(g,x)\mapsto (x,gx)$ is an isomorphism. Example 0.4 in Mumford's GIT is a variety $X$ with an action of $G=SL(2)$ with trivial stabilizers and geometric quotient $\mathbb{A}^1$, but such that $X$ is not a $G$-torsor over $\mathbb{A}^1$, so $[X/G]$ is not representable.