Let $A\subset B$ be two linear subspaces of a vector diace $V$ such that $\mathrm{dim}(V)=n$, $\mathrm{dim}(A)=p$, $\mathrm{dim}(B)=q$ and $\mathrm{dim}\left(\frac{V/A}{V/B}\right)=m$. Then what is the relation between $m,n,p,q$?
As is apparent from the question $m=q-p$, but is my guess right? If so can you please explain.
Yes, that's the only relation we can establish (besides $0\le p\le q\le n$).
We have $\dim V/A=n-p$ and $\dim V/B=n-q$. Their difference is just $q-p$.