Two monotone functions which equal on rational numbers

No. Take $$f(x)=x+\chi_{(\pi,+\infty)}(x)\,,\ \ \ g(x)=x+\chi_{[\pi,+\infty)}(x).$$ Here $\chi_A$ stands for the indicator function of the set $A$; i.e. $\chi_A$ is the function whose value at $x$ is $1$ if $x\in A$, and $0$ otherwise.