Four. At least known captures, and these were by British privateers. The Dutch were also engaged with the Spanish treasure fleet at times (Battles of La Naval de Manila), but it appears these actions resulted in the sinking , (but not the capture) of several ships in or near Manila (see article The Manila Galleons, by Steve Singer)
From the article on the Spanish Treasure Fleet:
In the case of the Manila galleons, only four were ever captured by
British warships in nearly three centuries: the Santa Anna by Thomas
Cavendish in 1589, the Encarnación in 1709 by Woodes Rogers, the
Covadonga by George Anson in 1743, and the Santísima Trinidad in 1762.
Two other British attempts were foiled by the Rosario in 1704 and the
Begonia in 1710.
The source article the Wikipedia entry quotes can be read here:
The Spanish Mariners: From the Discovery of America to Trafalgar. 1492-1805. Observations and Reflectionsby Paul V. Murray. This article agrees concerning the Dutch attempts:
The Dutch, the most implacable enemies the Spaniards faced, especially
in the first half of the seventeenth century, never managed to capture
a Galleon (though they attacked many) and never ever reached the walls
of Manila.
It is worth noting that the same source mentions that the last Manila galleon was actually raided of its silver by the Mexican revolutionaries in 1811, but I am not sure this should count, as the ship was still at shore:
In 1811, the Mexican insurgents took from a Galleon that was ready to
sail the silver cargo bound for Manila. The fortunes of war in those
years changed again; and with the Spanish in firm control of the
viceregal government, the same ship cast off for Manila in 1815, the
last of the great line ever to make the passage again.
Comments brought up the fact that Francis Drake took two Spanish treasure ships including the Nuestra Señora de la Concepción. If these 'count', then we are up to six recorded ships captured.
There appears to be some contradictory information available about this event, however. The Drake wikipedia entry says she was
sailing west towards Manila
The Concepción Wikipedia entry states she
was a 120-ton Spanish galleon that sailed the Peru–Panama trading
route during the 16th century.
There is no mention of Manila on that page at all. Another source,
Sir Francis Drake and the Famous Voyage, 1577-1580: Essays Commemorating the Quadricentennial of Drake's Circumnavigation of the Earth, agrees with the Panama route :
...he found no significant quantity of silver. A big consignment , he
was told, had recently left for Panama in the ship Nuestra Señora de
la Concepción
There is record, however, of a Nuestra Señora de la Concepción that was involved as a Manila Galleon, which went down in 1638 near Saipan. Some info on that ship can be found here:
On September 20, 1638, the Nuestra Senora de la Concepcion, a Spanish
galleon plying the lucrative trade route between Manila in the
Philippines and Acapulco, foundered in bad weather and was hurled onto
a reef. Most of the 400 people on board perished, and her precious
cargo from the Orient spilled into the sea.
There is a little more information concerning the ship:
The Concepcion was the largest Spanish ship built up to her time --
between 140 and 160 feet long and displacing some 2,000 tons, with a
loaded draft of between 18 and 22 feet.
I'm not an expert, however it seems that these are two different ships. This may account for the wikipedia mention of Drakes prize being bound for Manila. I do not think the ship/s captured by Drake would 'count' towards Manila Galleons for the purpose of this question.
(Sorry for the long tangent, but the Conception issue is almost a question on its own.)
