Well the most obvious way pagan gods were incorporated into Byzantine culture was that they were very visibly still there. If we take for example the ancient Athenian statue of Athena Promachos, crafted by Phidias for the acropolis, Konstantinos the Great removed the statue to Konstantinople in 330 and it was displayed there with hundreds of other examples of high cultural achievement until 1204. As such, knowing of these pieces of art and their heritage was a feature of educated elite Byzantine culture and they often feature in Niketas Choniates and Anna Komnene. In their texts, individuals will complete things with the grace of Tuxi/Fortuna and the Homeric ideas of divinely inspired action never leave elite literature. See Nikephoros Bryennios firing his bow like the Aphrodite-inspired Paris at Troy.

What is, to my mind at least, more interesting is how the lower classes perceived the hippodrome and capital's forums full of pagan monuments. One of the most interesting sources for 10th c Konstantinople is the Patria, lit. fatherland, which is an anonymous collection of foundation myths for the parts of the capital city. It is almost entirely inaccurate in terms of origin but tells us huge amounts about how the Christianisation of the empire had affected cultural memory. By 1204, the date of the greatest destruction of the ancient world during the fourth crusade, the people turned against the Athena Promachos as they saw it gesturing West and believed it to be welcoming the Italian and French crusaders and tore it down. The elites knew what the statue represented but to the lower classes they ahd destroyed a pagan artifact which in a time of crisis was fair game idolatry to be purged as a matter of zealotry.

There are other examples but this should get you started.