The Holy Spirit is God's Spirit and is mentioned in the Qur'an in many places.
Sura Yusuf 12:87, "Do not despair of Allah's spirit; none but unbelievers despair of Allah's spirit."
Sura al-Baqara 2:87 and 253, "We gave Jesus the son of Mary veritable signs and strengthened Him with the Holy Spirit."
Al Imam al-Nasafi said, "By the Holy Spirit is meant, the sanctified spirit...or the name of God the greatest."
Sura al-Ma'ida 5:110, "Jesus, son of Mary remember the favour I have bestowed on you and on your mother; how I have strengthened you with the Holy Spirit."
Al Sayyid Abdul Karim al-Djabali said about the Holy Spirit that He is not created, and what is not created is eternal and the eternal is God alone.
Also Al-Shaikh Muhammad al-Harira al-Bayyumi said, "The Holy Spirit is the Spirit of God and the Spirit of God is not created."
This is the holy Trinity in one God in whom we believe, and this is the secret of naming it as the Father, the Son, and the Holy Spirit.
The Father is the title of the essential Fatherhood of God.
The Son is the title of the incarnated Word of God.
The Spirit is the very Spirit of God.