In the first place, the Greek word καιροὶ = times, plural. The text of Luke 21 does not identify when these "times" were to begin. However, we have some very good clues based on the use of the plural word καιροὶ.
- Matt 16:3 - and in the morning, ‘Today it will be stormy, for the sky is red and overcast.’ You know how to interpret the appearance of the sky, but you cannot interpret the signs of the times.
- Acts 1:7 - Jesus replied, “It is not for you to know times or seasons that the Father has fixed by His own authority.
- Acts 14:17 - Yet He has not left Himself without testimony to His goodness: He gives you rain from heaven and fruitful seasons, filling your hearts with food and gladness.”
- Acts 17:26 - From one man He made every nation of men, that they should inhabit the whole earth; and He determined their appointed times and the boundaries of their lands.
- Eph 1:10 - to be put into effect when the times reach their fulfillment—to bring unity to all things in heaven and on earth under Christ.
- 1 Thess 5:1 - Now about the times and seasons, brothers, we do not need to write to you.
- Rev 12:14 - But the woman was given two wings of a great eagle to fly from the presence of the serpent to her place in the wilderness, where she was nourished for a time, and times, and half a time. [= 3½ years, see Dan 7:25, 12:7]
- Rev 11:2, 3 - But exclude the courtyard outside the temple. Do not measure it, because it has been given over to the gentiles, and they will trample the holy city for 42 months. And I will empower my two witnesses, and they will prophesy for 1,260 days, clothed in sackcloth.”
Thus, it appears "times of the Gentiles were:
- was future in Jesus' time
- was associated with 1260 days, 42 month, 3½ years as noted above.
Note the comments of the Cambridge commentary:
shall be trodden dawn of the Gentiles So that the very thing happened which the Maccabees had tried to avert by their
fortifications (1Ma 4:60). All sorts of Gentiles—Romans, Saracens,
Persians, Franks, Norsemen, Turks—have ‘trodden down’ Jerusalem since
then. The estai patounmene of the original implies a more permanent
result than the simple future. Comp. Revelation 11:2.
The question about the interpretation of the 1260 days, etc, is very vexed and disputed and should be the subject of a separate question.