I have not had this issue, and I have a 2nd generation Q, but from reading the board I'll suggest:
1) The received wisdom here is that you'll want to keep your tank at least half full while you are diagnosing this problem. The theory is that the pump is better cooled when it is immersed in gas.
2) As a minimum, it would be worth measuring the fuel pressure. If you want to go beyond the mere minimum, the more technically savvy on the board would suggest that you splice a T fitting into the rubber fuel hose downstream of the fuel filter and snake a few feet of fuel line hose from the T to the windshield, where you can tape a fuel pressure gauge so that you or your trusty assistant can read it while the car is in motion. If you do this, be sure you use fuel line hose and that you route it where it won't get cut by anything, or else you'll be driving a fireball down the interstate, and it's brown trousers time when that happens. If I were measuring fuel pressure that way, I'd keep the windows and the sunroof closed.
3) If I were you, I'd change the fuel filter...it's cheap and easy, it's not going to make anything worse, and it may relieve some strain on the system while you are diagnosing and dealing with other issues.
4) In addition to the fuel pump, there is also a "Fuel Pump Controller Unit". It is referred to on the board as FPCU. I suggest that you search on the terms "FPCU" "bees" "buzz" and "buzzing" (in addition to more obvious searches like "fuel pump")...those searches should lead you to posts about diagnosing the fuel pump and the FPCU. The received wisdom here is that if you ignore a bad fuel pump long enough, it will start to draw too much current, and damage the FPCU, which costs more than the fuel pump does.
5) You have downloaded the factory service manual, right?
6) If you conclude that it is your fuel pump, here's a write up a member did that is applicable to your year:
http://www.nicoclub.com/articles.php?id=164324
Good luck!