Mark 1-3

They went to Capernaum, and when the Sabbath came, Jesus went into the synagogue and began to teach. The people were amazed at his teaching, because he taught them as one who had authority, not as the teachers of the law. Just then a man in their synagogue who was possessed by an impure spirit cried out, “What do you want with us, Jesus of Nazareth? Have you come to destroy us? I know who you are—the Holy One of God!”

“Be quiet!” said Jesus sternly. “Come out of him!” The impure spirit shook the man violently and came out of him with a shriek.

That evening after sunset the people brought to Jesus all the sick and demon-possessed. The whole town gathered at the door, and Jesus healed many who had various diseases. He also drove out many demons, but he would not let the demons speak because they knew who he was. (Mark 1:21-26,32-34)

It's a little mysterious to me how demons know Jesus' identity and why Jesus prevents demons from speaking for that reason. Maybe he is teaching his followers something. Perhaps he is hoping to prolong his preaching and healing ministries, because the sooner he is publicly identified as Messiah, the sooner he will be opposed by the political and religious "establishment."

When the time is right Jesus accepts that his followers identify him as Messiah, reference Peter's claim, "You are the Christ, the Son of the Living God." (Matthew 16:16) For whatever reason here he is not ready for demons to reveal that information.