I am a bit confused about Theorem 2.6 in McCleary's "A USer Guide to Spectral Sequence". The theorem says that
Each filtered dg module $(A,d,F^*)$ (with decreasing filtration and $\deg(d)=1$) gives a spectral sequence $\{E^{*,*}_r, d_r\}$ of cohomological type (i.e., $\deg(d_r) = (r,1-r)$) with $$E^{p,q}_1 \cong H^{p+q}(F^p A / F^{p+1} A)$$
But what is the $(p+q)$-th cohomology? Since the differential respect the filtration, we should have the cochain complex $$\cdots \to F^p A^{q-1} / F^{p+1} A^{q-1} \to F^p A^q / F^{p+1} A^q \to F^p A^{q+1} / F^{p+1} A^{q+1} \to \cdots$$ induced by $d$. Then we can just take the cohomology of this cochain complex. Is this the $(p+q)$-th cohomology given in the theorem?
In the cochain complex in your second display, because you have the "cdots" symbol $\cdots$ on both the left and the right, it's not entirely clear to me how that sequence is indexed. Perhaps the superscript on $A$ is the index?
If so (and if that typo I suggested is corrected) then, just following the notation, $H^{p+q}(F^pA/F^{p+1}A)$ certainly does mean the $p+q^{\text{th}}$ cohomology of this cochain complex, which would be $$\frac{\text{Ker}\bigl(F^{p} A^{p+q} / F^{p+1} A^{p+q} \to F^{p} A^{p+q+1} / F^{p+1} A^{p+q+1}\bigr)}{\text{Image}\bigl(F^{p} A^{p+q-1} / F^{p+1} A^{p+q-1} \to F^{p} A^{p+q} / F^{p+1} A^{p+q}\bigr)} $$ However, I don't have that book in front of me so I cannot be sure this is correct. I would check the book carefully to see exactly which is the $0$-dimensional term of the cochain complex $F^p A / F^{p+1} A$, which the $1$-dimensional term, etc.