Lewis & Short

Q, q, the sixteenth letter of the Latin alphabet (in which i and j were reckoned as one), concerning the origin of which the ancients were in doubt, some correctly supposing it to be the Greek Koppa
(Note:), transferred from the Dorian alphabet of Cumæ, Quint. 1, 4, 9; Ter. Maur. p. 2253 P.; Mar. Victor. p. 2459 and 2468 ib.; while others erroneously explained it as a mere graphical contraction of C and V, Vel. Long. p. 2218 P.; Ter. Maur. p. 2399 ib.; cf. Diom. p. 420 ib.; Mart. Cap. 3, § 255; Isid. Orig. 1, 4, 14. There is a perpetual vacillation between the spelling cu, q, and qu in the inscrr. and MSS.; hence q frequently stands for c. In early inscriptions, PEQVDES and PEQVNIA occur for pecudes and pecunia (Lex Thor. lin. 14 and 19); QVM for the prep. cum, Inscr. Vet. ap. Orell. 566, and also upon a coin, A. U. C. 737; and QVOM for the prep. cum, in the fourth epitaph of the Scipios, and in the Lex Thor. lin. 21: QVOQIRCA for quocirca in the Lex Jul. Municip.: IN OQVOLTOD for in occulto, S. C. Bacch. On the other hand, for quod stands CVOD, Inscr. Orell. 3882; for aquae, ACVAE, Inscr. Grut. 593, 5. But qu before a u sound does not occur during the Republican period, when quom or cum, equos, locuntur, anticus, etc., were the forms in use; v. Rib. prol. Verg. p. 442 sq.; 449; Brambach, p. 20 sq.
On the vacillation of the oldest MSS. between cu and qu, see Freund ad Cic. Mil. p. 31 sq.
Q often corresponds with the Greek π: Lat. quinque, equos, sequor; Gr. πέντε (πέμπε) ἵππος, ἕπω.
And also with the Gr. τ, for which the Oscan has p: Gr. τίς, τί; Oscan pis, pit; Lat. quis, quid: Gr. τέ; Oscan pe; Lat. que: Gr. τέτταρα; Oscan petora; Lat. quattuor; on the origin of the Lat. qu in an lndo-European kv, v. Corss. Ausspr. 1, 67 sqq.; Ascoli, Vergl. Lautl. 1, p. 49 sqq.; cf., on the development of qu from c in the Latin language itself, Corss. Ausspr. 2, 356 sq.
As an abbreviation, Q designates most freq. the prænomen Quintus, but also stands for Quaestor, que, quinquennalis, al. Q. I. S. S. quae infra scripta sunt. Q. R. C. F. quando rex comitiavit fas. Q. S. P. P. S. qui sacris publicis praesto sunt. Q. V. A. qui vixit annos. S. P. Q. R. senatus populusque Romanus, etc.