Such a bound is impossible in general. E.g., suppose that $\Omega=[0,1]$, $g=0$, and $f=a^{-1/p}1_{[0,a]}$, where $p>0$ and $a\downarrow0$. Then for any $q\in(0,p)$ we have $\int|f-g|^q=a^{1-q/p}\to0$, whereas $\int|f-g|^p=\int|(f^p-g^p)=1\not\to0$.

Such a bound is impossible in general. E.g., suppose that $\Omega=[0,1]$, $g=0$, and $f=a^{-1/p}1_{[0,a]}$, where $p>0$ and $a\downarrow0$. Then for any $q\in(0,p)$ we have $\int\lvert f-g\rvert^q=a^{1-q/p}\to0$, whereas $\int\lvert f-g\rvert^p=\int(f^p-g^p)=1\not\to0$.