From the Historia Brittonum
In the meantime, Vortigern, as if desirous of adding to the evils he had already occasioned, married his own daughter, by whom he had a son. When this was made known to St. Germanus, he came, with all the British clergy, to reprove him: and whilst a numerous assembly of the ecclesiasts and laity were in consultation, the weak king ordered his daughter to appear before them, and in the presence of all to present her son to St. Germanus, and declare that he (*) was the father of the child. The immodest woman obeyed; and St. Germanus, taking the child said, "I will be a father to you, my son; nor will I dismiss you till a razor, scissors, and comb, are given to me, and it is allowed you to give them to your carnal father." The child obeyed St. Germanus, and going to his father Vortigern, said to him, "You are my father; shave and cut the hair of my head." The king blushed, and was silent; and, without replying to the child, arose in great anger, and fled from the presence of St. Germanus, execrated and condemned by the whole synod.
(*)The 'he' being Germanus
Historically speaking, no one really believes this tale. There is no mention of any allegations of incest against Vortigern by Gildas in his De Excidio Britanniae, and Gildas was not one to miss the chance to accuse a native British ruler of any sin, real or imagined. Neither does he mention any confrontation between Germanus and Vortigern, and since Gildas is our only vaguely contemporary record, we can safely dismiss the tale to the realms of fantasy.
The allegation of immorality in one's enemies is always very convenient and the tale was probably invented by the Anglo-Saxons and enthusiastically adopted by Bede (who recounts it with a certain relish) as a pseudo-justification for their interventions in Britain. It became just as convenient for the Medieval Welsh to blame the loss of dominion over the island of Britain one one immoral king.