Where did Jesus' belief in the devil come from?

It depends on your definition of 'Old Testament.' If you mean the Catholic or Orthodox/pre-Reformation Old Testament, then you have quite a few Old Testament references to the devil or devils, whereas the Books contained in the Protestant canon do not so much mention them.

For example,

Tobit 6:14, 16 (DRB)

Then Tobias answered, and said: I hear that she hath been given to seven husbands, and they all died: moreover I have heard, that a devil killed them. ...

Then the angel Raphael said to him: Hear me, and I will shew thee who they are, over whom the devil can prevail.

Tobit 12:14 (DRB):

And now the Lord hath sent me to heal thee, and to deliver Sara thy son's wife from the devil.

(Tobit tells of events that transpire in 800 B.C., though since we humor 'scholars,' is at least 200 B.C. as to the date it was penned.)

Or,

Wisdom 2:23-24 (DRB)

For God created man incorruptible, and to the image of his own likeness he made him. But by the envy of the devil, death came into the world: and they follow him that are of his side.

(Wisdom is generally dated around 100 BC. give or take.)

This intertestamental belief on the devil fits nicely with e.g. Matthew 4:1, and the rest of the development in the New Testament on the person of Satan:

Matthew 4:1 (DRB)

Then Jesus was led by the spirit into the desert, to be tempted by the devil.

(Matthew is dated mid first century A.D.)

Revelation 12:9 (DRB)

And that great dragon was cast out, that old serpent, who is called the devil and Satan, who seduceth the whole world; and he was cast unto the earth, and his angels were thrown down with him.

(Revelation is generally dated to around 90 A.D.)

Keep in mind Jesus was speaking in 'the Old Testament period' during the early part of His ministry. St. John the Baptist is technically an Old Testament prophet in addition to the herald of the very presence of Christ.

But besides these more explicit references (not that if the Bible didn't mention a word about 'the devil' this wasn't what God's people believed!), the devil is clearly what is meant by 'the Accuser' or 'Satan' in the so-called proto-canonical Books of the Old Testament—even if a 'proper' use of 'the devil' would only later become more normative.

For example, we read in Job that Satan is undoubtedly 'smug:'

Job 1:9-11 (DRB)

And Satan answering [God], said: Doth Job fear God in vain ? Hast not thou made a fence for him, and his house, and all his substance round about, blessed the works of his hands, and his possession hath increased on the earth? 11 But stretch forth thy hand a little, and touch all that he hath, and see if he blesseth thee not to thy face.

Or one of the more explicit references to the disobedient and sinful a nature of this angel called the Accuser.

Zechariah 3:1-2 (DRB)

And the Lord shewed me [Joshua] the high priest standing before the angel of the Lord: and Satan stood on his right hand to be his adversary. 2 And the Lord said to Satan: The Lord rebuke thee, O Satan: and the Lord that chose Jerusalem rebuke thee: Is not this a brand plucked out of the fire?

Compare also with,

Jude 1:9 (DRB)

When Michael the archangel, disputing with the devil, contended about the body of Moses, he durst not bring against him the judgment of railing speech, but said: The Lord [rebuke] thee.

In Judaism, there most certainly is a devil, first alluded to in Genesis 3. The serpent mentioned that deceived Eve is known to be the devil, and we can specifically see a more direct reference to the devil itself, named a garden cherub, being cast from "the holy mountain of God" starting in Ezekiel 28:15.

Further, the story in the book of Job has its premise in a direct conversation of God with Satan. We know Jesus had read Job, because he was wise in the Scriptures, and the Hebrew writer (someone contemporary with the birth of New Testament Christianity) referenced Job in Hebrews 11:4.

Most definitively, we have the encounter of Jesus with the devil directly as he was tempted in the desert for 40 days. Jesus knows he is the Son of God, and the devil knows it, too, and tries to get him to bow before him, but of course he fails, and leaves upon hearing "Begone, Satan!" in Matt. 4:10.

The Jews of Jesus' time also spoke of demons frequently, even going as far as to question if Jesus' miraculous casting out of demons was by the power of 'the prince of demons', Beelzebub/Beelzebul (Matt. 12:27) (which of course, it was not, because it was done by the authority of God.).

Those are just some of the tangible ways we can see that Jesus could have known about the devil, not the least of which was in Scriptural records and oral tradition. However, do not forget that as Christians, we believe that Jesus is holy and descended from heaven itself, and had all spiritual knowledge needed to teach the world what it needed to know. The demons (sometimes called 'unclean spirits') I mentioned previously also were familiar with Jesus, just as Jesus is familiar with them and the devil they serve, demonstrated in Acts 19:15 when the sons of Sceva unsuccessfully tried to exorcise a demon. The demon, upon hearing the command to come out of a man in the name of 'the Jesus that Paul proclaims', replied "Jesus I know, and Paul I recognize, but who are you?".

In short, the Bible is filled with references to Satan, both in the Old and New testaments. The Old testament, which Jesus evident ally studied much of, references the devil plenty. I am not familiar with the modern practice of Judaism, but in ancient times they still had more than just the Torah, they had the books of the prophets (especially Isaiah). The new testament, that was written concerning Jesus, references interactions with the devil and demons frequently, as proof that Jesus was familiar with, interacted with, and had power over them.

I hope my answer was clear enough.