There was a line from the DS9 pilot "Emissary" in which Sisko and Dax knew where they had emerged from the Bajoran Wormhole in the Gamma Quadrant because an earlier Federation probe had mapped that area and transmitted its findings back to the Federation.
To nitpick, we didn't learn that it would have been a Federation probe, and we didn't learn that it would have actually visited that location. We could be speaking of an ancient mission whose findings were available to the Feds, or of long-base interferometry studies made possible by the use of probing spacecraft sent just a few thousand lightyears away from each other - or of a mission involving no spacecraft as such, and referred to as a "probe" simply because it probed into the Gamma Quadrant with good-quality telescopes.
We get more definite information from VOY "Friendship One", where a warp spacecraft from Earth is very explicitly quoted with the ability to cross a quarter of the galaxy within no more than 300 years, and possibly within 170 years already. This would support the view that the Federation (especially thanks to its more advanced early members) had plenty of capability for charting spots everywhere in the galaxy and beyond.
Whether those spots put together would add up to 19% of the galaxy is a wholly different matter. A mission covering 70,000 ly will only create a thin thread of explored space between the endpoints; spacecraft sensors only reach across a dozen lightyears even in the most extreme cases. We can't assume that every galactic probe would have been carrying a device on par with the Argus Array, when even the big manned starships have nothing comparable aboard.
Then there's this map from "The Chase":
Doesn't necessarily tell us much about the Federation, though: Professor Galen would probably have been showing Picard a map of ancient civilizations, considering his profession. Heck, he's the sort who'd use a map of ancient civilizations to find his way to a conference on Bolarus, or a vacation on Risa! But knowledge of these civilizations would be plausible if the UFP had knowledge of the grids shown in "The Emissary", as the grids would at least overlap if not quite cover these mysterious amber outlines.