No, Muhammad is not mentioned or predicted in the Bible.
Some Muslims quote Deut. 18:15-18 and say it refers to Muhammad; here Moses, says:
The Lord your God will raise up for you a prophet like me from among you, from your brothers—it is to him you shall listen— … And the Lord said to me, … I will raise up for them a prophet like you from among their brothers. And I will put my words in his mouth, and he shall speak to them all that I command him.
God and Moses are speaking to the Israelites, the Jews. They say that there will be a prophet in the future who would be like Moses and who would be ‘from among you [the Israelites]’ — i.e. the prophet would be a Jew. This shows that Deut.18 isn’t referring to Muhammad because Muhammad wasn’t a Jew, he was an Arab.
The New Testament shows who Deut. 18:15-18 is really talking about: it is a prophecy about Jesus. Acts 3:17-26 quotes Deut.18:18 (in v22) and applies it to Jesus:
{17} “And now, brothers, I know that you acted in ignorance, as did also your rulers. {18} But what God foretold by the mouth of all the prophets, that his Christ would suffer, he thus fulfilled. {19} Repent therefore, and turn again, that your sins may be blotted out, {20} that times of refreshing may come from the presence of the Lord, and that he may send the Christ appointed for you, Jesus, {21} whom heaven must receive until the time for restoring all the things about which God spoke by the mouth of his holy prophets long ago. {22} Moses said, ‘The Lord God will raise up for you a prophet like me from your brothers. You shall listen to him in whatever he tells you. {23} And it shall be that every soul who does not listen to that prophet shall be destroyed from the people.’ {24} And all the prophets who have spoken, from Samuel and those who came after him, also proclaimed these days. {25} You are the sons of the prophets and of the covenant that God made with your fathers, saying to Abraham, ‘And in your offspring shall all the families of the earth be blessed.’ {26} God, having raised up his servant, sent him to you first, to bless you by turning every one of you from your wickedness.”
Stephen, in Acts 7:37,52, also tells us that Deut.18:15-18 is referring to Jesus:
{37} This is the Moses who said to the Israelites, ‘God will raise up for you a prophet like me from your brothers.’ […] {52} … the prophets … announced beforehand the coming of the Righteous One, whom you have now betrayed and murdered,
Jesus, not Muhammad, was the Jewish prophet who was like Moses.
Some Muslims also quote Jesus’ words in John 14-16 about the ‘Helper’ (ESV; ‘Comforter’ — KJV; ‘Counselor’ — NIV; Greek: Parakletos) and say they refer to Muhammad; e.g., in John 14:16 Jesus says:
And I will ask the Father, and he will give you another Helper, to be with you forever,
First, notice that Jesus says the ‘Helper’ would be with the twelve disciples. However, Muhammad was not born until 570 AD, long after the twelve disciples were dead: Muhammad wasn’t with the twelve disciples, so the ‘Helper’ does not refer to Muhammad.
Also notice that Jesus says the ‘Helper’ would be with the disciples ‘forever’. However, Muhammad died in 632 AD, so he wasn’t with the disciples ‘forever’. Muhammad cannot be the ‘Helper’.
Even more decisively, a few verses further on in John 14 Jesus tells us who the ‘Helper’ is; he says (John 14:26):
… the Helper, the Holy Spirit, whom the Father will send in my name, he will teach you all things and bring to your remembrance all that I have said to you.
Jesus tells us the ‘Helper’ is the Holy Spirit,* which was given to the disciples in Acts 2 (see Acts 1:8; 2:1-4).
So, the Bible does not mention or predict Muhammad.
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*For more on this, see Who or what is the Advocate or Comforter?