From a couple of sources it seems that they didn't originate the idea of Avodah Zara but assisted in developing it further.
First, on the development of idolatry, Rambam states that idolatry came from man in a gradual process (Hilchot Avodah Zara 1:1):
"During the times of Enosh, mankind made a great mistake, and the wise men of that generation gave thoughtless counsel. Enosh himself was one of those who erred.
Their mistake was as follows: They said God created stars and spheres with which to control the world. He placed them on high and treated them with honor, making them servants who minister before Him. Accordingly, it is fitting to praise and glorify them and to treat them with honor. [They perceived] this to be the will of God, blessed be He, that they magnify and honor those whom He magnified and honored, just as a king desires that the servants who stand before him be honored. Indeed, doing so is an expression of honor to the king.
After conceiving of this notion, they began to construct temples to the stars and offer sacrifices to them. They would praise and glorify them with words, and prostrate themselves before them, because by doing so, they would - according to their false conception - be fulfilling the will of God.
This was the essence of the worship of false gods, and this was the rationale of those who worshiped them. They would not say that there is no other god except for this star."
From here we see that mankind was responsible for the creation of idolatry.
Midrash of Shemchazzai and Azael:
"R. Joseph was once asked what was the story of Shemḥazai and Azael, and he replied, 'When the generation of Enosh arose and worshipped idols, and when the generation of the flood arose and went astray, God was grieved that He had created man, as it is said, "And the Lord repented that He had made man, and He was grieved at heart." Then two angels, whose names were Shemḥazai and ‘Azael, appeared before God, and said, "O Lord of the universe, did we not say unto Thee when Thou didst create Thy world, 'Do not create man'?" as it is said, "What is man, that Thou shouldst remember him?" "Then what shall become of the world?" said God. They replied, "We will occupy ourselves with it." God said, "It is revealed and well known to Me that if peradventure you had lived in that earthly world, the evil inclination would have swayed you just as much as it rules over the sons of man, but you would be more stubborn than they." "Give us Thy sanction, then, and let us descend among the creatures, and then Thou shalt see how we shall sanctify Thy name." "Descend," spake the Lord, "and dwell ye among them."
Forthwith He allowed the evil inclination to sway them. As soon as they descended and beheld the daughters of man that they were beautiful, they began to disport themselves with them, as it is said, "When the sons of Elohim saw the daughters of man," they could not restrain their inclination...When Shemḥazai and ‘Azael saw this they took to them wives, and begat children. The former begat two children, whose names were Heyya, and Aheyya. And ‘Azael was appointed chief over all the dyes, and over all kinds of ornaments by which women entice men to thoughts of sin..."
Me'am Lo'ez Bamidbar, pp. 271-272 (my translation):
"And why was that place called 'eyes unveiled'? For there were two angels who spoke negatively about the creation of man and their names were Azah and Azael. And when the generation of the flood came along, which was a generation that rebelled and worshipped idolatry and did many sins...and when they [the angels] descended, they saw beautiful women and their evil inclination overpowered them and they could not control their inclination and sinned with them. And then the Holy One Blessed is He saw that these angels were sinning and using the names of impurity that they knew and were causing mankind to sin even more..."
From these two sources we see that the angels didn't teach idolatry, they just taught things that made it even worse - dyes and ornaments to entice women and names of impurity. Idolatry had already begun to be developed during the time of Enosh (and see Tanchuma Noach 18:3).
It should be noted that there are traditions about the Nefilim in apocryphal literature and in the Dead Sea Scrolls (together called 'Enochic literature') and there are several parallels between these sources and traditional Jewish sources. For example, also according to 1 Enoch ch. 7-8, the angels didn't teach idolatry. They taught witchcraft, how to make weapons from metal, and usage of dyes, ornaments and gemstones.
