When did the Ottoman Turks begin incursions into the Byzantine empire?

Answer 1

Turkish tribes began incursions to the Byzantine Empire around 1300 AD. These peoples came from Central Asia and began to see opportunities in capturing parts of the Byzantine Empire. This process continued until the Turks captured Constantinople in 1453.

Answer 2

Background
Oğuz Turks, of which the Ottomans were only one tribe (albeit the most successful in the later period), began attacking the Byzantines as soon as the Battle of Manzikert in 1071. In that battle the Seljuk Turks (an allied Oğuz Turkic tribe) decisively defeated the Byzantines, opening up central and eastern Anatolia to Oğuz Turkish invasions. The Ottoman tribes only began to form a state-let in the 13th century and began their attacks against the Byzantines only when the much larger power had its gaze towards Italy.

Specific Answer
Osman I, the founder of the Ottoman Empire (and its namesake), began the Ottoman raids into Byzantine-controlled Anatolia with the cities of Sogut and Eskisehir falling to Ottoman Empire in 1265 and 1289 respectively. The Ottomans would eventually put an end to the Byzantine Empire with the conquest of Constantinople in 1453.