Yes, the Holy Spirit indwelled believers before Christ was born. Romans 8:9 says,

But ye are not in the flesh, but in the Spirit, if so be that the Spirit of God dwell in you. Now if any man have not the Spirit of Christ, he is none of his.

According to this verse, believers living prior to the birth of Jesus must have had the Spirit of God dwelling in them because they belonged to God.

Isaiah 63:11 says,

Then he remembered the days of old, Moses, and his people, saying, Where is he that brought them up out of the sea with the shepherd of his flock? where is he that put his holy Spirit within him?

Ezekiel 36:27 says,

And I will put my spirit within you, and cause you to walk in my statutes, and ye shall keep my judgments, and do them.

Ezekiel 3:24 says,

Then the spirit entered into me, and set me upon my feet, and spake with me, and said unto me, Go, shut thyself within thine house.

Elihu says in Job 32:18

For I am full of matter, the spirit within me constraineth me.

Exodus 31:3 says,

And I have filled him with the spirit of God, in wisdom, and in understanding, and in knowledge, and in all manner of workmanship.

Exodus 35:31 says,

And he hath filled him with the spirit of God, in wisdom, in understanding, and in knowledge, and in all manner of workmanship.

Prior to the birth of Jesus Christ, Luke 1:67 says,

And his [John’s] father Zacharias was filled with the Holy Ghost, and prophesied.

The Scriptures confirm that those who lived before Jesus was born were filled and indwelt with the Holy Spirit of God.