Does the term 'Holy Spirit' as used by the Christians refer to Allah?

All perfect praise be to Allah, the Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.

Christians believe in the doctrine of the Trinity. In brief, the Trinity means that there are three hypostases (three Divine Persons); namely, the Father, the Son, and the Holy Spirit. They believe that God is three persons in one divine essence. Such was what they claimed.

Al-Qurtubi wrote:

"Christian sects unanimously believe in the doctrine of the Trinity. They say that God is one essence in three consubstantial persons (or hypostases). Each hypostasis is a god. They mean by the hypostases: Existence, Life, and Knowledge. They may refer to the hypostases as the Father, the Son, and the Holy Spirit. By the Father, they mean Existence; by the Holy Spirit, they mean Life; and by the Son, they mean Jesus Christ. This is part of their views which involve much confusion over their most fundamental beliefs."

Al-Mawsooʻah Al-Muyassarah Lil-Mathaahib wa Al-Adyaan Al-Muʻaasirah (Simplified Encyclopedia of Contemporary Beliefs and Religions) says about the Christian meaning of the Holy Spirit and hypostases:

"The Holy Spirit: The Holy Spirit which came upon Mary with glad tidings (at the Annunciation), came upon Jesus in the form of a dove at his baptism,  came upon the apostles after Jesus was raised, and which still exists and continues to descend upon the priests and saints at the church, guiding, teaching, and bestowing spiritual blessings and bounties on them is the spirit and life of God; a true deity from a true deity.

The Hypostases: ... and it is for this reason that they believe in the three hypostases—the Father, the Son, and the Holy Spirit—in what they claim to be 'one God in three Divine Persons and three Divine Persons in one God'. That is a plainly false claim that they themselves found hard to comprehend, so they differed dramatically regarding its meaning. Each of their sects accused the others of disbelief because of it. Allah, the Exalted, declared all of them to be disbelievers if they do not desist from what they are saying; He says (what means): {They have certainly disbelieved who say, 'Allah is the third of three.' And there is no god except one God. And if they do not desist from what they are saying, there will surely afflict the disbelievers among them a painful punishment.} [Quran 5:73]"

Allah knows best.