Stack Exchange is a global site. I'm sure that while a majority of the user base is in the Western Hemisphere time zone, I'm also sure that there is a wide distribution of when answerers are online and active - on their lunch break, after dinner, after the 11 PM news, etc. You also need to be careful about assuming that a good answer will posted at or around the time the question was posted. I often mull a question for a while before posting and if you think about that logically that should be true as well. Also when I am most active is not always the time that I have available to formulate answers. I often read a bunch of questions and note the ones I want to think about and come back and ask later.
I don't know that there's any way to draw anyManishEarth drew some relevant statistics from data.se, but "better" is quite subjective. Just because an answer to a question posted at 1100 UTC got 15 up-votes while an answer to a similar question posted on a different day at 1800 UTC only got 5 up-votes does not mean that the former was a "better" answer. And more answers does not mean better answers. Now, I don't know if they keep statistics on active volume / sessions vs. individual questions and answers - I'm sure they have some of that information somewhere, I just don't know if it is exposed to us.
If you tend to participate in a few tags frequently, you may have noticed users that often produce high quality answers. So one thing you may wish to do (on a much smaller scale than your question implies) is to observe when they post most of their answers. Then you at least have a slightly higher chance at getting answers from people you know give good answers. Keep in mind though that their answers posted at 1800 UTC might be on a question they first read at 1100 UTC.
Possibly more relevant
One thing I have observed as having an affect on the quality of answers is the quality of the question. If you ask a poor question, you get poor answers. Time of day is irrelevant. And regardless of the quality of the question, on SO at least, you get no shortage of answers in either case. More on that in a minute.
There is also the "rush" syndrome. People seem to be obsessed with posting the first answer (for badges like Enlighted, aka FGITW), or as early as possible (for a better chance at a share of the up-votes).
One other issue is the badgering I often see for the OP to accept someone's answer. We took away the accept rate in part to alleviate this badgering (which, admittedly, did not always). But still this can occasionally pressure OPs to accept answers before a large majority of the potentially interested population get to see them.
Now, the problem with all three of these behaviors as I see it is that much less attention is paid to questions that have answers. And this is particularly true for questions that have up-voted and, most importantly, accepted answers. But the folks most likely to spend time and effort crafting a good answer are likely to gravitate to questions where they are likely to have the greatest impact: those with few / no answers and certainly without an accepted answer. The theory about that is that if an answer has been accepted, the OP already has a solution and probably doesn't care much about new solutions, and if there are multiple answers, the competition might be harder.
On a site with millions of users, all questions do get additional views in spite of these facts. But I suspect that a lot of the factors that influence whether you get the best possible answer have less to do with time of day and more to do with dumb luck / coincidence.