Do we know exactly who they were?
Yes. A better question is if these are the people who wrote the four Gospels, and that's less clear.
Matthew refers to Jesus' apostle Matthew, also named Levi. He was a publican that Jesus called to give up his life as a tax collector and follow Him.
Luke refers to one of Paul's missionary companions.
Mark refers to John Mark, a missionary companion of both Paul and Peter.
John most likely refers to the Apostle John, like Matthew one of Jesus' Twelve Apostles.
The Gospel accounts themselves are anonymous, but these four individuals have been ascribed traditional authorship since very ancient times, the earliest possible mention comes to us from the late 1st century/early 2nd century church father Papias of Hierapolis. The writings of Papias are mostly lost, but fragments have been preserved in the writings of others, among those preserved fragments include Papias' account of who authored the four Gospels.
Papias' accounts,
"
Mark having become the interpreter of Peter, wrote down accurately whatsoever he remembered. It was not, however, in exact order that he related the sayings or deeds of Christ. For he neither heard the Lord nor accompanied Him. But afterwards, as I said, he accompanied Peter, who accommodated his instructions to the necessities [of his hearers], but with no intention of giving a regular narrative of the Lord's sayings. Wherefore Mark made no mistake in thus writing some things as he remembered them. For of one thing he took special care, not to omit anything he had heard, and not to put anything fictitious into the statements." and
"Matthew put together the oracles [of the Lord] in the Hebrew* language, and each one interpreted them as best he could." - Papias, Fragment VI, as recorded by Eusebius
Another major source is Irenaeus from the late 2nd century,
"
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. After their departure, Mark, the disciple and interpreter of Peter, did also hand down to us in writing what had been preached by Peter. Luke also, the companion of Paul, recorded in a book the Gospel preached by him. Afterwards, John, the disciple of the Lord, who also had leaned upon His breast, did himself publish a Gospel during his residence at Ephesus in Asia." - Irenaeus, Against Heresies, Book 3, Ch. 1
*Aramaic, as Aramaic was often called "Hebrew" in antiquity, since it was the language of the Hebrews i.e. the Jewish people. The patristic view is that Matthew originally composed a list of Jesus' sayings (
logion, meaning "sayings" or "words" but translated here as "oracles") in Aramaic, but later wrote the Greek text we know today.
Many modern scholars doubt these traditions, so it's an open question as to who wrote the Gospels since they are anonymous.
But the individuals mentioned are well known, even if they aren't the ones who penned the Gospels traditionally ascribed to them.
-CryptoLutheran
Fig 34 St Huberts RC Church Dunsop Bridge Lancs by Prince Arthur, on Flickr
