No authoritative answer to this question can be given, because New Testament authors did not explicitly mark all the places where they quoted or alluded to the Old Testament. Indeed, many of them probably would not have been able to do so even if they tried, since they had internalized the Old Testament text so thoroughly, and thus they adopted its language as their own.
As a result, researchers have broken down New Testament uses of the OT into categories like "direct quotation," "indirect quotation," "allusion," and "possible allusion," and, not surprisingly, they disagree on what fits into each category. So in the end, no two lists are the same.
That said, there are still some helpful resources that begin to tackle the question. The first one I'll mention is a free project from the Blue Letter Bible, "Parallel Passages in New Testament Quoted from Old Testament," which has the following goal:
The list contains not only the direct or indirect citations, but also the allusions which are particularly worthy of attention
Thus it will not necessarily contain all the quotes/allusions, just ones considered particularly noteworthy. A quick analysis of this list gives the following totals:
- Quotes: 302
- Allusions: 493
- Possible allusions: 138
- Total: 933
The second resource, a more in-depth one, is the book Commentary on the New Testament Use of the Old Testament, edited by Beale and Carson. It works through the entire New Testament, calling out many parallels between the NT and OT texts. Depending on your locale, you may be able to preview parts of this work for free on Google Books.
One final caveat: Besides the aforementioned difficulty in compiling such a list, both of the resources mentioned here come from Protestant sources, and thus will not typically take note of references to any deuterocanonical books recognized as part of the Old Testament by Catholics and Eastern Orthodox. Thus if such references are also desired, these sources will be incomplete in that respect as well.