I think one of the oldest historical mentions of Jesus is by Josephus [1][2]. There is, however, scholarly discussion about whether parts of his references to Jesus were altered by later Christian scribes [3].
The traditional view held by Christians is that the Gospel of Matthew was written within 10 years of Jesus' death. Modern scholars (often atheists) do not believe it though.
Do you have a source for this? I've never heard anyone claim that they written so early. For Catholics at least, I think it's a point that the Church and some Tradition are older than the Gospels, i.e. that the Gospels are written by the Church, for the Church (not the other way around)
Catholics traditionally held Matthew was written shortly after Jesus' death (there is some debate about the actual date). Many modern Catholics, accept the modern scholarship nowadays though. There are traditions created prior to the gospels regardless of the dating. I would also mention the Bible explicitly states there are non-written traditions in 2 Thessalonians 2:15.
Irenaeus said "Matthew also issued a written Gospel among the Hebrews in their own dialect while Peter and Paul were preaching at Rome and laying the foundations of the church."
There is some debate on Peter's death, but it is usually placed in 64-68 AD. Paul died either 64 or 65 ad. This is prior to when modern scholars state.
I don't have a quote, but Eusebius said he thought it was written about 12 years after Jesus' death.
There are plenty of other early Christians who took the position that the gospels were prior to 70 AD, but I am on my phone and don't want to try to find them. I was looking for a Catholic one that I previously saw that said it was 5 years, but couldn't find it.
As far as I can tell, every prominent Christian until the 1800s thought they were earlier than modern scholars say.
The reason I am using the 70 AD date is because of the temple being destroyed then. One of the arguments modern scholars use is that Jesus prophesized that the temple was going to be destroyed. Since God does not exist, Jesus could not have known that and as such it had to be written after 70 ad.
[1] https://en.wikipedia.org/wiki/Josephus
[2] https://en.wikipedia.org/wiki/Josephus_on_Jesus
[3] https://scholar.google.com/scholar?hl=en&as_sdt=0%2C5&q=jose...