Gould's statement is (at a minimum) hyperbole, since there is no single composer or even a single historical period which one can justifiably call “the end of modality” or “the beginning of tonality”. The practices which we lump together as “common-practice tonality” emerged from a very long historical process of change that arguably begins in the 14th century, if not earlier, and of course continued in the nineteenth and twentieth centuries. Nor is it really obvious any more that there is a clear opposition between two different systems called “modality” and “tonality”.
That said, Gibbons (1583–1625) lived during a period – the end of the 16th and beginning of the 17th century – which is often associated historically with the beginning of certain “modern” ways of thinking about harmony. More typically, these are associated with his Italian contemporaries – most notably Monteverdi – and with the newly “chordal” way of conceiving music brought about by the beginnings of basso continuo (figured bass), as well as the new harmonic practices permitted by the seconda prattica of madrigal and opera.
As an English composer primarily of polyphonic music, Gibbons has little direct connection to these new practices. Yet there are some features of his music which connect naturally to later “tonal” ways of organising harmony.
Several Gibbons fantasies organise their points of imitation into a circle-of-fifths sequence, so that successive entrances bring in the same melody successively in different “keys” – for instance, successively “in A major”, “in D major”, “in G major”, etc. Examples occur in the keyboard fantasy in G minor, the fantasy for “double organ”, and at least two of the three-part consort fantasies. Circle-of-fifths motion is, of course, a very common technique in the later Baroque. The more standard use of imitation would be to create tonal stability through entrances on only two notes, usually the first and fifth scale degrees, although there are plenty of earlier precedents for Gibbons's technique: for example, there are motets by Palestrina with similar sequentially imitative sections. As usual, it is difficult to point to a single composer or composition as the definitive origin.
Some people have also remarked that Gibbons's use of the major mode in his C- and G-final pieces seems subjectively very close to our expectations for “C major” and “G major”. Examples of this would be the keyboard prelude in G and the two keyboard fantasies in C. The prelude in G, in particular, is another piece in which the circle of fifths dominates, in this case because of the continual succession of cadences rather than an imitative scheme. Whether Gibbons's practice in these pieces is really more “modern” or “tonal” in its orientation than that of his predecessors and contemporaries is difficult to say with certainty, but it is an impression that several people besides Gould have expressed. (I apologise for not having a citation to hand).
If you can get access to JSTOR through a library or university, you might be interested to read Candace Bailey's paper on Orlando Gibbons, Keyboard Music and the Beginning of the Baroque, which discusses the keyboard prelude in G in particular. She also writes that Gibbons's music was copied into manuscripts disproportionately often in the years after his death by other English organists, which may suggest that they, too, found something more “forward-looking” about his music than that of his immediate contemporaries.