I can't give chapter and verse, but I'm pretty sure that in some episode someone says something like "we've only explored 25% of the Alpha Quadrant".
I can't find the citation, but I believe Janeway once lamented that the Alpha Quadrant was almost completely explored. (The context was a "look on the bright side" kind of thing: she was getting to do far more exploring than she would have if she hadn't gotten sent to the Delta Quadrant.)
On TNG, in season one (Where No One Has Gone Before) Kosinsky says "In three centuries of space flight, we've charted just eleven percent of our galaxy."
In season two (The Dauphin), Wesley says "We've only charted nineteen percent of our galaxy". Given that the Alpha Quadrant represents 25% of the galaxy, if the Federation has explored most of it, the vast majority of that 19% must lie within the Alpha, and it is the Beta Quadrant that is curiously unexplored.
* It is possible to reconcile these two figures without saying that the Federation explored 8% of the galaxy in one year, as Kosinsky seemed to be talking specificly about
human exploration, while Wesley seemed to be talking about not even just the Federation, meaning that about 58% of the territory charted by the Federation was explored by the Federation, while 42% is territory of one of the Federation's neighbors who shared their maps, but no human has actually gone there yet.