The answer is no. The Holy Spirit is another name for God.
In John's gospel we read this account of Jesus' baptism:
John 1:29-34
29 The next day he saw Jesus coming to him, and said, "Behold, the Lamb of God who takes away the sin of the world! 30 "This is He on behalf of whom I said, 'After me comes a Man who has a higher rank than I, for He existed before me.' 31 "And I did not recognize Him, but in order that He might be manifested to Israel, I came baptizing in water." 32 And John bore witness saying, "I have beheld the Spirit descending as a dove out of heaven, and He remained upon Him. 33 "And I did not recognize Him, but He who sent me to baptize in water said to me, 'He upon whom you see the Spirit descending and remaining upon Him, this is the one who baptizes in the Holy Spirit.' 34 "And I have seen, and have borne witness that this is the Son of God." NAS
Now jump to John 14.
John 14:9-10
9 Jesus said to him, "Have I been so long with you, and yet you have not come to know Me, Philip? He who has seen Me has seen the Father; how do you say, 'Show us the Father'? 10 "Do you not believe that I am in the Father, and the Father is in Me? The words that I say to you I do not speak on My own initiative, but the Father abiding in Me does His works. NAS
Now read Jesus prayer in John 17. Who is conspicuous by His absence in this prayer? The Holy Spirit is not mentioned, however Jesus repeats that the Father is in Him and He is in the Father.
John 17:20-21
20 "I do not ask in behalf of these alone, but for those also who believe in Me through their word; 21 that they may all be one; even as Thou, Father, art in Me, and I in Thee, that they also may be in Us; that the world may believe that Thou didst send Me.
NAS
Even if you want to believe that John 17 is a private conversation between the Father and the Son, it is inconceivable that this conversation does not even hint at the promised Helper or Comforter Jesus promised to send in John 14:16, 26, 15:26, and 16:7.