Presumably $F$ and $G$ are both right-continuous with-left limits functions of local finite variation in an interval that contains $[a,b]$. Lebesgue integration by parts states that
Theorem: Let $F$, $G$ be right--continuous functions of locally finite variation on an interval $I$ (bound or unbounded) Let $\mu_F$ and $\mu_G$ the Stieltjes-Lebesgue measures generated by $F$ and $G$ respectively. For any bounded interval $(a,b]\subset I$,
\begin{align}
\int_{(a,b]}F(t)\mu_G(dt)=F(b)G(b)-F(a)G(a)-\int_{(a,b]}G(t-)\mu_F(dt)\tag{0}\label{zero}
\end{align}
where $G(t-)=\lim_{s\nearrow t}G(s)$.
It is common to use $dF(x)$ and $dG(x)$ instead of $\mu_F(dx)$ and $\mu_G(dx)$ respectively. Using measure notation is clearer to me.
Now, Notice that for any singleton $\{x\}$, $\Delta G(x):=\mu_G(\{x\})=G(x)-G(x-)$, $\Delta F(x)=\mu_F(\{x\})=F(x)-F(x-)$. Then,
\begin{align}
\int_{(a,b]}G(t-)\mu_F(dt)&=\int_{(a,b]}(G(t)-\Delta G(t))\mu_F(dt)\\
&= \int_{(a,b]}G(t)\mu_F(dt)- \int_{(a,b]}\Delta G(t)\mu_F(dt)\\
&=\int_{(a,b]}G(t)\mu_F(dt)-\sum_{a<t\leq b}\Delta G(t)\Delta F(t)
\end{align}
Notice that as $F$ and $G$ are of finite local variation, each function $F$ and $G$ has at most a countable set of discontinuities in $(a,b]$ and that $\sum_{a<t\leq b}|\Delta G(t)\Delta F(t)|<\infty$. Consequently
\begin{align}
\int_{(a,b]}F(t)\mu_G(dt)&=F(b)G(b)-F(a)G(a)-\int_{(a,b]}G(t)\mu_F(dt)+\sum_{a<t\leq b}\Delta G(t)\Delta F(t)\tag{1}\label{one}
\end{align}
If $G$ and $F$ do not have common jumps (i.e. point of discontinuity) the. the last term in the right-hand side of \eqref{one} is $0$.
