There are some problems in your question. $H$ usually refers to the enthalpy, defined as $$H\equiv U+PV,$$
where $U$ is internal energy, $P$ is pressure, and $V$ is volume. The Helmholtz energy, also called the Helmholtz free energy or Helmholtz potential, is often denoted by $F$ and is defined as $$F\equiv U-TS,$$ where $T$ is temperature and $S$ is entropy.
In addition, the Gibbs free energy $G$ is defined as $$G\equiv U+PV-TS.$$ With all these varying parameters, partial derivatives must indicate which parameters must stay constant; thus, we could write $$\left(\frac{\partial G}{\partial T}\right)_{U,P,V}=\left(\frac{\partial G}{\partial T}\right)_{V,N}=-S,$$ for example, but removing the information about which variables to hold constant would make the expression nearly meaningless.
Note carefully the differences from your question.
Now, as you write, the specific latent heat $q_i$ (i.e., the latent heat per unit mass) can be defined as $mq_i=H_\mathrm{high~T}-H_\mathrm{low~T}$. At a phase transition at constant temperature and pressure, the fundamental relation tells us that $$\require{cancel}dG=-S\,\cancelto{0}{dT}+V\,\cancelto{0}{dP}+\sum_i \mu_i N_i=\sum_i \mu_i N_i$$ with $N_\mathrm{high~T}=-N_\mathrm{low~T}$ from conservation of matter, indicating that the partial Gibbs free energies $\mu_i$ of the two phases are equal and that the Gibbs free energy is constant: $$\Delta G=G_\mathrm{high~T}-G_\mathrm{low~T}=0;$$ from this, we have $$H_\mathrm{high~T}-H_\mathrm{low~T}-T(S_\mathrm{high~T}-S_\mathrm{low~T})=0,$$ yielding $$H_\mathrm{high~T}-H_\mathrm{low~T}=T(S_\mathrm{high~T}-S_\mathrm{low~T}),$$ QED. Does this make sense?
