In which one of the four Gospels (Matthew, Mark, Luke, John) is the Old Testament quoted the most?
The answer to this question depends somewhat on the definition of "quoted," but we can draw some high-level conclusions. BlueLetterBible.org provides a list of New Testament citations and allusions to the Old Testament, and an examination of that dataset reveals the following:
Book Direct/indirect citations Total citations & allusions Matthew 44 (49) 77 (102) Mark 18 (23) 30 (39) Luke 20 (24) 52 (68) John 14 (17) 31 (49)
The first number is the number of unique NT references to OT passage(s). The number in parentheses is the total number of cited OT passages (since a single NT verse can cite more than one OT verse). No matter how you count it, however, Matthew is the clear winner, given BLB's definition of citations and allusions.
Matthew's dominance is also confirmed by commentators. Craig L. Blomberg, writing in the Commentary on the New Testament Use of the Old Testament, writes:
The Hebrew Scriptures [...] permeate Matthew's Gospel. Approximately fifty-five references prove close enough in wording for commentators typically to label them "quotations," compared to about sixty-five for the other three canonical Gospels put together. [...] In addition to explicit quotations, numerous allusions and echoes of Scripture may be discerned in every part of this Gospel, roughly twice as often as in Mark, Luke or John.
Blomberg's numbers vary somewhat from BLB's, but demonstrate that even with varying definitions of what a "quote" or "allusion" is, Matthew refers to the Old Testament significantly more than the other Gospel writers.