22 Then came the Festival of Dedication at Jerusalem. It was winter, 23 and Jesus was in the temple courts walking in Solomon’s Colonnade. 24 The Jews who were there gathered around him, saying, “How long will
you keep us in suspense? If you are the Messiah, tell us plainly.”
Does the "Festival of Dedication" refer to Hanukkah? If so, does this mean that Jesus was present at the temple to celebrate Hanukkah?
The "festival of dedication" (or "feast") in this verse was indeed the forerunner of what is today Hanukkah. It was established by Judas Maccabeus in the intertestamental period, as Jamieson, Fausset, and Brown write:
[This festival of the Dedication] was instituted by Jude Maccabeus, to commemorate the purification of the temple from the profanations to which it had been subjected by Antiochus Epiphanes 165 B.C., and kept for eight days, from the twenty-fifth Chisleu (December), the day on which Judas began the first joyous celebration of it (1 Maccabees 4:52,56,59; and Josephus, Antiquities, 7.7.7).
Now, the festival goes by the name Hanukkah, as Wikipedia indicates, but it is observed on the same day and celebrates the same event.
As to whether Jesus actually celebrated Hanukkah, that's a bit more controversial. John Wesley, following Grotius, thinks so:
So our Lord observed festivals even of human appointment. Is it not, at least, innocent for us to do the same? (source)
But Calvin suggests that his purpose was to further his ministry:
Christ appeared in the temple at that time, according to custom, that his preaching might yield more abundant fruit amidst a large assembly of men. (source)
So yes, this is a reference to Hanukkah, but it's harder to say if Jesus actually "celebrated" the holiday.
The feast of the dedication was a festival celebrating the re-dedication if the sacrificial altar. It was instituted by by Judas Maccabaees. This was later referred to as the festival of lights. It was/is Hanukkah.
On the lunar-solar calendar this celebration falls in December and sometimes in November.
While this is somewhat opinion, there is no reason to believe as a Jew that Christ wouldn't participate in this important festival.
All the feasts point to Christ, and we know 100% that Christ celebrated the feasts.
Likewise, Paul celebrated the feasts, even after the resurrection - and he said that the Feasts are a shadow of things to come, but the substance is of Christ.
Here follows three reasons we know that Christ celebrated the feasts:
Because He created them, and God instructed his children to keep them forever, and Jesus always did what the Father said. In fact, the Jews were sent into exile for 70 years, because they did not keep the Jubilee years.
We have record of Christ celebrating the Passover with his disciples.
In the passage of Jesus at the transfiguration, Peter said let us build 3 tabernacles, and the word in the original Greek, is the same word for Feast of Booths- Sukkot. These booths are temporary shelters, which was a reminder of when they wandered in the wilderness.
We would expect to find this in the Old Testament in Leviticus, where God established the 7 feasts. The reason it's not here, is that this is the dedication of the 2nd temple, which happened in the 400 year period between the Testaments, so we see it in the New Testament.
To answer your question directly - Yes, this passage you mention from John 10:22 doesn't just refer to Hanukkah - in the original language, the Hebrew word for Feast of Dedication actually is "Hanukkah", and several translations actually say Hanukkah.
John 10:22-24 Jesus Claims to Be the Son of God
22 It was now winter, and Jesus was in Jerusalem at the time of Hanukkah, the Festival of Dedication.
23 He was in the Temple, walking through the section known as Solomon’s Colonnade.
24 The people surrounded him and asked, “How long are you going to keep us in suspense? If you are the Messiah, tell us plainly.”
25 Jesus replied, “I have already told you, and you don’t believe me. The proof is the work I do in my Father’s name.
26 But you don’t believe me because you are not my sheep.
27 My sheep listen to my voice; I know them, and they follow me.
28 I give them eternal life, and they will never perish. No one can snatch them away from me, 29 for my Father has given them to me, and he is more powerful than anyone else. No one can snatch them from the Father’s hand.
30 The Father and I are one.”
The other reason we know that Christ celebrated the feasts is that His second advent is inseparably linked with the 3 Fall feasts. When his disciples asked about the end of the world and when he would return, he used a Jewish idiom, that was instantly understood and recognizable to them. He said he would return when "no one knows the day or hour" which was unmistakable reference to Feast of Trumpets. It is the only feast that happens at the New Moon, and sometimes it's rainy or cloudy and the sliver of the new moon cannot be seen. This feast begins the Jewish Civil calendar and all 7 feasts and all the weekly Sabbaths are determined by the 2 witnesses who see the New Moon, and then the New Year begins. When the new moon is sighted by the two witnesses in Jerusalem, there are 99 alternating long and short blasts on the shofar, and then the 100th blast is the longest of all, and it is known as "the Last Trumpet".