I'm reasonably sure that this story can only be found in Nikephoros' Short History, which was written in the late eighth century by the patriarch of Constantinople. It sounds pretty unlikely and I think this was invented by later Byzantine writers to condemn Cyrus further - he had been damned as a heretic at the Ecumenical Council of 680, so doing this only blackened his name further. Cyrus got a terrible reputation since he was hated by both miaphysite and Chalcedonian Christians, as I discussed here, so I wouldn't really trust Nicephoros' story.

Still, it is true that Cyrus negotiated with Amr and seemingly had a cordial relationship with him, as evident in the seventh-century Chronicle of John of Nikiu and the eighth-century Chronicle of Theophilus of Edessa. Bribery, or rather the paying of tribute, was a very common policy for the Roman Empire, so that part sounds entirely reasonably, especially as this is backed up by other sources. The conversion and marriage part, rather less so, as this is attested to in no other source. Pseudo-Sebeos writing from Armenia in the 660s did mention that the Arab army in Egypt during the First Fitna converted to Christianity, so I guess we can make a very shaky case for it, but it sounds very unlikely.