The same goes for Descent - they didn't say which Quadrant they were in.

The borg aren't just in the Delta Quadrant - I would have to say that they spill over into the beta Quadrant as well... or that they have sparse areas at the fringes of their space... i.e. they wouldn't have cubes patrolling empty space for no reason... and that Q might have blown the Enterprise to the nearest Borg ship which was in the Beta Quadrant...

P.S. That map was made all the way back in season 1. Whoever has touched the map up assinging new planets and systems to random stars... the map might have been updated at one time to include 'Ferenginar' for the Art of Star Trek, but there are a whole heap like the Breen Homeworld that someone has just plonked in there. I'd take this map with a grain of salt - it is basically a cool looking map that just was a nice link to TOS from season 1 TNG showing all the planets that the E-Nil visited - I would take it with a grain of salt, considering what we know about now how the Romulans, Klingons, Cardassians, the Federation, The location of cestus III and the involvment of the Tholians into Federation/Klingon/Romulan affairs is known... when it wasn't known back in TNG season 1.

As for the "veracity" of that map... in the TNG
Companion, Larry Nemecek mentions an "art department map" which the employees apparently kept updating. In the entry for "Suddenly Human", he mentions that the Talarians were to the "west" on that map. Curiously, none of the published or leaked or even retouched versions of this "Conspiracy" map have shown the location of Talar. So that art department map may have been something different altogether.

Somehow, I doubt that, though. This map was seen in many, many places and in many, many forms. Not only was it a wall display of the above format in a couple of episodes. It was also part of a galactic "top view", a big unlabeled display screen seen in for example "The Emissary". Furthermore, it was seen as part of a similar but different "top view" behind the back of an Admiral Picard was teleconferencing with in "Data's Day". And it apparently served as inspiration for the map Dr Galen used in "The Chase", although the details do not quite match.

Before that, a nitpicking fan would probably have assumed that Borg home turf is in Beta, since that's the direction in which Wolf 359 lies.

A fun connection: originally, the menace of the Borg was supposed to be the same as the menace of the parasites from "Conspiracy", until writing decisions took a twist or two. And in "Conspiracy", we learn the parasites had been "clearing a corridor" by influencing Starfleet ships and installations in a specific direction. That direction included starbase 12, which is in the Beta quadrant as well (it's the one closest to Pollux, as per "Who Mourns for Adonais?", and we all know Pollux is in Beta...)!

I remember feeling, and understanding that this was a general consensus, that the Borg were in the Delta Quadrant, and that Voyager was going to run into them.

No, because as has been already pointed out, the quadrant system wasn't even invented until season 3 of TNG. It first appeared in "The Price" (and ST VI).

The Borg were not mentioned as being from the Delta Quadrant out loud at any point during TNG. The reason we knew that the Borg were in the Delta Quadrant before Voyager said so it for the same reason we know what a Nebula-class ship is, and did know long before it was said out loud: we're internet nerds who listen when Okuda says something like "The Borg are in the Delta Quadrant on my maps, kiddies!"