How to improve the quality and visibility of research?
Red makes your glove shoot out fireballs, whereas pink heals you.
Amazingly broad question but a couple of simple answers.
Indexing like you suggested. Make sure the appropriate indexes exist and if the need is great enough make sure they are all covering.
Get rid of the *. Specify the columns you need. If you are pulling across 100 columns over a million rows that's going to be a LOT of data. If you only need 3 columns only specify 3 columns.
Change the timeout. This sounds hokey but honestly sometimes it really is the answer. SQL Server (for example) doesn't actually have a timeout so the problem is on the connection side. Have them increase the timeout to 30 seconds (assuming this is an acceptable amount of time, and it frequently won't be).
Make sure it's the DBMS's fault. It's very possible you are having a problem on the connection side that's causing it to take 5-10 seconds just to connect. Fix that and you are well within your time.
The result evidently follows only from absolute convergence of $Q(s)$. As I mentioned in the question, any product of two absolutely convergent Dirichlet series is an absolutely convergent Dirichlet series, so let
$$ \frac{Q^k(s)}{k!} = \sum_{n=1}^{\infty} \frac{a_{k,n}}{n^s}. $$
Then
$$ e^{Q(s)} = \sum_{k=0}^{\infty} \frac{Q^k(s)}{k!} = \sum_{k=0}^{\infty} \sum_{n=1}^{\infty} \frac{a_{k,n}}{n^s}. $$
If we assume $Q(s)$ converges absolutely, then the inner sum converges asbolutely (after which the outer sum clearly does too), so we may swap the order of summation to get
$$ e^{Q(s)} = \sum_{n=1}^{\infty} \sum_{k=0}^{\infty} \frac{a_{k,n}}{n^s} = \sum_{n=1}^{\infty} \frac{b_n}{n^s}, $$
where
$$ b_n = \sum_{k=0}^{\infty} a_{k,n}. $$
Conversely, if $\sum_n \frac{b_n}{n^s}$ converges absolutely, then, since the $a_{k,n}$ are all nonnegative, the double sum $\sum_n \sum_k \frac{a_{k,n}}{n^s}$ converges absolutely, and hence switching the order of summation allows us to conclude that each $\sum_n \frac{a_{k,n}}{n^s}$ converges absolutely. In particular, $Q(s)$ converges absolutely, as desired.