ho.history overview - What proof of quadratic reciprocity is Hilbert referring to in this quote?

It could be Kronecker's determination of the sign of the Gauss sum by means of Cauchy's theorem. Already Gauss noted that the determination of the sign implies the law of quadratic reciprocity.

In response to the request for references:

Leopold Kronecker: Summirung der Gauss'schen Reihen ... J. Reine Angew. Math. 105 (1889), 267-268.

Also in volume 4 of his Werke, 297-300. (This was where I xeroxed it, so I can vouch for the page numbers, I have the pages in front of me right now).

Also in Landau's Elementare Zahlentheorie (together with two others, by Mertens and Schur),
near the end of the book.

Also supposed to be in Ayoub: Introduction to the Analytic Theory of Numbers, but I am not familiar with his book, so I cannot vouch for this.

There is a later determination of the sign of the Gauss sum by contour integration, due to Mordell, which is quite accessible; it is in Chandrasekharan's Introduction to Analytic Number Theory, page 35--39. Chandrasekharan does a more general case.

Now, I have not claimed that Kronecker's proof was the one that Hilbert was thinking of. I cannot read the mind of a dead man (nor that of a living one).