Assume that $\,I\subseteq\mathbb{R}\,$ is an interval and $\,β:\,I\rightarrow\,I\,$ a strictly increasing and continuous function with a single fixed point $\,s_0\in I\,$, satisfying $\,(s_0-t)(β(t)-t)\leq 0\,$ for all $\,t\in I$, where the equality occurs only when $\,t=s_0$. Hamza et al. considered the general quantum operator, $\,D_β[f](t):=\displaystyle\frac{f\big(β(t)\big)-f(t)}{β(t)-t}\,$ when $\,t\neq s_0\,$ and $\,D_β[f](s_0):=f^{\prime}(s_0)\,$ when $\,t=s_0\,$. It generalizes the Jackson $\,q$-derivative operator $\,D_{q}\,$ as well as the Hahn (quantum derivative) operator, $\,D_{q,ω}$. We obtained Grüss type inequalities for its inverse operator, the $β$-integral. Furthermore, we introduced the concept of $\,β$-Riemann-Stieltjes integral and obtained Grüss type inequalities associated with it.