Can one win with two dark square bishops and two light square bishops and his opponent has only one queen? Can one win with two dark square bishops and two light square bishops and his opponent has only one queen?

I remember seeing this endgame being analyzed in a video around 10 years ago (and I think the answer is yes) but could not find the video any more.
What you're referring to is a so-called "7-man" ending (2 kings, 1 queen and 4 bishops). Luckily for you, we have 7-man tablebases which are even available online and which allow us to analyze every possible legal 7-man chess ending.

[FEN "1q1k4/8/8/8/B7/1B6/1B6/2B1K3 w - - 0 1"]


Randomly setting up the above position shows that white is indeed winning, and generally most positions seem to show white winning. However, this isn't always the case, for example the following is a draw (white to move) because of the fact that black can either fork the king and one of the bishops, or can fork two bishops:

[FEN "3k3K/B3q3/8/8/8/8/B7/2B4B w - - 0 1"]


The tablebase also shows how the win often doesn't require that many ply, depending on the starting position of course.