Let's say I have 3 groups for differential gene analysis. I want to make two pairwise comparisons, A vs B and B vs C. I have 1000 genes.

I can provide contrasts to edgeR/DESeq2 for each of the pairwise comparison, thus I will have two contrasts.

The first time I test A vs B, I will have a list of q-values for my 1000 genes. Then next time I run B vs C, I will have a new set of q-values for my 1000 genes.

Here's my question: the q-values are adjustment for my 1000 genes. However, I'm actually testing 2000 genes (A vs B and B vs C) simultaneously. Should I further adjust my q-values?

You could do an F-like-test for each gene, then you would only be doing 1000 tests. In edgeR, you simply give both pairwise comparisons to glmLRT() or glmQLFTest() simultaneously and it will do a combined test.

Alternatively, if you want to keep the two comparisons separate, see ?decideTests for the various ways in which you can further adjust the p-values across comparisons. Usually people just adjust each comparison separately.