I have some RNASeq data from a microbe that was grown either aerobically or anaerobically, and samples were taken at T0, T1, T2, or T3.

To set up a model matrix describing my sample, I have done the following...

permut <- model.matrix(~ 0 + aerobicity + timing)

And this yields the following, dropping level T0, which makes sense.

   aerobic anaerobic T1 T2 T3
1        1         0  0  0  0
2        1         0  0  0  0
3        1         0  1  0  0
4        1         0  1  0  0
5        1         0  0  1  0
6        1         0  0  1  0
7        1         0  0  0  1
8        1         0  0  0  1
9        1         0  0  0  0
10       1         0  0  0  0
11       1         0  1  0  0
12       1         0  1  0  0
13       1         0  0  1  0
14       1         0  0  1  0
15       1         0  0  0  1
16       1         0  0  0  1
17       0         1  0  0  0
18       0         1  0  0  0
19       0         1  1  0  0
20       0         1  1  0  0
21       0         1  0  1  0
22       0         1  0  1  0
23       0         1  0  0  1
24       0         1  0  0  1
25       0         1  0  0  0
26       0         1  0  0  0
27       0         1  1  0  0
28       0         1  1  0  0
29       0         1  0  1  0
30       0         1  0  1  0
31       0         1  0  0  1
32       0         1  0  0  1

When I set up contrasts, I can easily compare the aerobic to anaerobic growth conditions:

aerobicity_contrast <- makeContrasts(aerobic - anaerobic, levels = permut)

But, I am sad to say that I am stumped trying to set up a contrast between T3 and T0. Since the level is dropped, I am not sure how to extract the relevant contrasts from the model. The following clearly doesn't work:

timing_contrast <- makeContrasts(T3 - T0, levels = permut)

However, dropping T0 from the makeContrasts works, but I am unsure if this is appropriate given the model.

timing_contrast <- makeContrasts(T3, levels = permut)

Any advice would be appreciated.

Your comment is correct, but there is actually no need for a contrast at all to test T3 vs T0. The T3 coefficient is already the T3 vs T0 contrast.

That's why glmQLFTest and glmLRT have a coef argument as well as a contrast argument. Just set coef="T3" to test the T3 vs T0 contrast.