The New Testament doesn’t merely quote the Old Testament – the New Testament relies entirely on the Old Testament for its meaning. I’ve listed below authors that will help you see this.
Closely related to this subject is seeing Christ in the Old Testament – a narrower and more specific focus on the same subject.
G. K. Beale and D. A. Carson
C. H. Dodd (1884-1973) – Wrote the classic According to the Scriptures: The Sub-structure of New Testament Theology (1952), and, in the same year, the pamphlet “The Old Testament in the New.” Dodd and his book were cited as an important influence by Richard Hays in the introduction of Reading Backwards.
E. Earle Ellis (1926-2010) – Wrote Paul’s Use of the Old Testament (1957). He wrote other books touching on this subject, but most were academic in nature and not easy to find.
Leonhard Goppelt (1911-1973) – Wrote Typos: The Typological Interpretation of the Old Testament in the New (1939).
Richard B. Hays (1948-) – Hays has written extensively on this subject, often using terms of literary criticism such as “intertextuality” (his B.A. was in English Literature). Beginning with with Echoes of Scripture in the Letters of Paul (1989), he went on to write The Conversion of the Imagination: Paul as Interpreter of Israel’s Scripture (2005), and, most recently, Reading Backwards: Figural Christology and the Fourfold Gospel Witness (2014). In the preface of this last book, he credits as influences for his approach to this subject C. H. Dodd, Barnabas Lindars, Nils Dahl, and Donald Juel.