Is $H_m - H_n$ a surjection onto $\mathbb{Q}^+$?

You cannot get $H_m-H_n=2$, or any integer $\ge2$.

In a sum $1/(n+1)+\cdots+1/m$ exactly one of the terms has minimal $2$-adic valuation. This can only be zero if one has only one term. Therefore there's a power of $2$ in the denominator, which cannot be $2^0$ save in the cases $H_{n+1}-H_n=1/(n+1)$ where $n+1$ is odd.

The proof given at ProofWiki and at Is there an elementary proof that $\sum \limits_{k=1}^n \frac1k$ is never an integer? for the fact that no harmonic number $H_n$ for $n\gt1$ is an integer also works for this case.

There is a unique denominator in $[a,b]$ that is divisible by the highest power of $2$. If $q$ is an integer, multiply through by one less factor of $2$ to turn all terms except one into integers (which is impossible). The only case to which this doesn’t apply is when the sum only runs over a single odd integer; and in this case the sum is also not an integer, except for $H_1=1$.