Laudator Temporis Acti

Kathleen Norris, Dakota: A Spiritual Geography (Boston: Houghton Mifflin, 1993), p. 140:

I know that Thomas Jefferson, who first read Plato's Republic in Greek at the age of seventy-one and found it overrated, believed that the independent farmer was a foundation stone of American democracy. But, knowing that the words for liberty and library come from the same Latin root, he also believed that the farmer had to be well read for democracy to work.

I can't verify the claim that Jefferson first read Plato's Republic at age 71, although I find it stated several places on the World Wide Web. I checked the indices to all of the volumes of Dumas Malone's biography of Jefferson, and the name Plato does not appear.

But there seems to be no etymological connection between the Latin adjective līber (free) and the Latin noun lĭber (book). See the Online Etymological Dictionary's entries on liberal and library:

liberal (adj.)
c.1375, from O.Fr. liberal "befitting free men, noble, generous," from L. liberalis "noble, generous," lit. "pertaining to a free man," from liber "free," from PIE base *leudheros (cf. Gk. eleutheros "free"), probably originally "belonging to the people" (though the precise semantic development is obscure), from *leudho- "people" (cf. O.C.S. ljudu, Lith. liaudis, O.E. leod, Ger. Leute "nation, people").

library
c.1374, from Anglo-Fr. librarie, from O.Fr. librairie "collection of books," noun use of adj. librarius "concerning books," from L. librarium "chest for books," from liber (gen. libri) "book, paper, parchment," originally "the inner bark of trees," probably a derivative of PIE base *leub(h)- "to strip, to peel" (see leaf). The equivalent word in most Romance languages now means "bookseller's shop."