Did exactly 3 wise men travel on camels to visit Jesus in the manger?

Matthew 2:1-12 records the visit of the wise men. Out of this story, all kinds of popular imagery has developed which actually has nothing to do with Matthew's account.

First: Scripture doesn't say how many wise men there were. The notion that there were three of them arises from the fact that they presented three gifts. There could have been any number of men.

Second: Their mode of transportation is nowhere mentioned. It's very likely they used camels in their travels, but it isn't stated.

Third: The wise men did not visit Jesus in the manger. This is an unfortunate fallacy that people pick up from nativity scenes depicting both the shepherds and the wise men paying homage to the newborn Jesus together. Perhaps that makes for pleasing imagery to some, but it's biblically inaccurate.

According to Luke's gospel, angels announced Christ's arrival to shepherds in the field on the night of His birth, and theyimmediately went and saw a "babe" (Greek brephos - "a new-born child") in the manger (Luke 2:8-15). Luke doesn't tell of the visit of the wise men.

Conversely, Matthew's account doesn't tell of the shepherds' visit; indeed, it says nothing of events on the night of Jesus' birth, and some time elapses between the close of Matthew 1 (the naming of Jesus, which would officially have taken place at His circumcision, when He was eight days old - Luke 2:26) and the beginning of Matthew 2.

When the wise men arrived, Matthew 2:11 says that Jesus was a "young child" (Greek paidion - "a young child, a little boy, a little girl; of a more advanced child"), and that they found Him in a "house" with Mary His mother (not in the manger). By this time, Jesus would have been up to two years old, as evidenced by Herod's subsequent "slaughter of the innocents:" all male children in the region aged two and under (Matthew 2:16).

To summarize the answer to your question:

No, The Bible doesn't say there were "exactly" three wise men.

Matthew's account makes no mention of camels.

The wise men didn't visit the infant Jesus in the manger; they visited Him as a "young child" in a "house."