Have a read of the John Romero interview from Quake Expo 2006 (the page is gone now, but you can still find it on the internet archive). From what he says, Trent's soundtrack (and sound effects) was pretty much the last thing that made it into the game - the fact that the test demo has placeholder sound effects seems to corroborate this, so if anything I'd say Trent was influenced by what he saw in the game, not the other way around.
I've seen in a few interviews from id that Quake generally wasn't a fun game for them to make - according to Romero (in his interview with Matt Barton), American McGee disappeared when crunch time came. Tonnes of time was spent being creatively frustrated while waiting for John Carmack to get the engine into a usable state and with a deadline/the public opinion looming the whole time.
If anything, I think a lot of the texture and tone of the game seems to come from the idea that well, everyone making the game was miserable and tensions were about to snap (and they indeed did once the game actually dropped, resulting in Romero's departure). It's actually my favourite id game from an emotional standpoint - in my interpretation, all the frustration and misery of its developers was being channelled into the game's creation and it seeps through every aspect of the visuals and gameplay.
It's so refreshing to see a video on Quake that isn't preoccupied with the whole multiplayer/Quakeworld side of things for a change and actually explores the artistic and tonal component of the game, since that's what still resonates with me 18 years after the fact. Happy new year!