The best approach to word studies is the synchronic method; that is, one should consider what terms possibly mean at a certain time--not what they potentially denote through history. Finding the "exact meaning" of a word can be extremely challenging or elusive, but fortunately there are entries in the major NT lexica for terms, including the word θεόπνευστος.
BAGD supplies us an example from Plutarch: THS QEOPNESTOU SOFIHSLOGOS ESTIN ARISTOS. It also recommends the thorough work by B.B. Warfield "Revelation and Inspiration" 27, 229-259. Lastly, BAGD lists 2 Tim. 3:16 and then provides "inspired by God" for θεόπνευστος.
Nevertheless, one caveat needs to be made. Warfield cites the source this way:
In the unique passage, Plutarch, de placit. phil., 5, 2 (Mor. 904, 2): ‘tous oneirous tous theopneustous kat anagken ginesthai’. ‘Tous de phusikous aneidolopoioumenes psuches to sumtheron aute ktl’., it is very probably to be ascribed to the copyist, and stands, as Wyttenbach conjectures, in the place of ‘theopemptous’. Besides this it occurs in Pseudo- Phocylides, v. 121: ‘tes de theopneustou sophies logos estin aristos’— unless the whole line is, with Bernays, to be deleted as disturbing to the sense—as well as in the fifth book of the “Sibyllines,” v. 308: ‘Kume d he mora sun namasi tois theopneustois’, and v. 406, ‘Alla megan genetera theon panton theopneuston En thusiais egerairon kai hagias ekatombas’. The Psuedo-Phocylides was, however, a Hellenist, and the author of the fifth book of the “Sibyllines” was, most probably, an Egyptian Jew living in the time of Hadrian. On Christian ground we find it in II Tim. iii. 16, which is possibly the earliest written employment of it to which we can point.
I need to verify the exact wording myself, but the main point of this missive is what θεόπνευστος possibly denotes. See https://www.the-highway.com/God-Inspired-Scripture_Warfield.html
Louw-Nida has: "θεόπνευστος, ον: pertaining to a communication which has been inspired by God--inspired by God, divinely inspired" (Semantic Domain 33.261). Ralph Earle writes that the phrase "Given by inspiration" (KJV) is "one word in Greek, QEOPNEUSTOS (only here in NT). It literally means 'God-breathed'--QEOS, 'god,' and PNEW, 'breathe.' That is, God breathed His truth into the hearts and minds of the writers of Scripture. The best translation is 'God-breathed' " (NIV).
Yet Louw-Nida warn us that a dictational view of θεόπνευστος should not be adopted (33.261). So we must acknowledge that God apparently did not dictate his thoughts to men. Furthermore, as we study the Bible, we find that concepts are not limited to words: one should not confuse Wort with Begriff. For example, the book of Hosea hardly uses the word (or related terms) "love." Yet the Divine acts depicted therein give evidence of Jehovah's love even if the prophet does not employ the exact term. Linguistically speaking, one word can be used to express many different concepts (polysemy) and one concept can be expressed by many different words (synonymy or overlapping relations).
Applying this principle to θεόπνευστος, it can be observed that while the word does not mean "infallible and inerrant," we are seemingly justified inferring that when God speaks--He speaks utterances that do not contain error and untruth. If we compare 2 Pet. 1:20, 21 with 2 Tim. 3:16, we may conclude that θεόπνευστος suggests infallibility and inerrancy. As the psalmist writes: "The sayings of Jehovah are pure sayings, As silver refined in a smelting furnace of earth, clarified seven times" (Ps. 12:6, 7).