The 18-year-old joined the Hammers on a free transfer from Chelsea two years ago but has only made six senior appearances since, though he has been fruitful for the Under-23s.
On transfer deadline day, Quina was believed to have travelled to the Hornets' training ground, with negotiations taking place over a switch to the Hertfordshire club.
Although the window slammed shut at 5pm, Watford were reported to have been granted a two-hour extension by the Premier League.
According to Sky Sports News, the deal has now been completed, though neither club has yet made an announcement.
The Portugal youth international, whose contract expires next season, has also been linked with Barcelona.