Muḥammad has absolutely no role in Christianity and is never referred to in the Christian Bible.
The first παράκλητος ("Comforter") was Yeshu'a the Anointed One ("Jesus Christ").
In 1 John 2:1, it is written,
My little children, I write these things to you so that you do not sin. And if anyone sins, we have a comforter (παράκλητον) with the Father, Jesus Christ the righteous.
Τεκνία μου, ταῦτα γράφω ὑμῖν, ἵνα μὴ ἁμάρτητε καὶ ἐάν τις ἁμάρτῃ, παράκλητον ἔχομεν πρὸς τὸν πατέρα, Ἰησοῦν Χριστὸν δίκαιον
Furthermore, Yeshu'a mentions ἄλλον παράκλητον ("another Comforter").
In John 14:16-17, it is written,
16 And I shall ask the Father, and He shall give you another Comforter (ἄλλον παράκλητον), so that he may remain with you forever, 17 the Spirit of truth, whom the world cannot receive, because it does not see him, neither does it know him, but you know him, for he dwells with you, and he shall be in you.
16 καὶ ἐγὼ ἐρωτήσω τὸν πατέρα καὶ ἄλλον παράκλητον δώσει ὑμῖν ἵνα μένῃ μεθ᾽ ὑμῶν εἰς τὸν αἰῶνα 17 τὸ πνεῦμα τῆς ἀληθείας ὃ ὁ κόσμος οὐ δύναται λαβεῖν ὅτι οὐ θεωρεῖ αὐτὸ οὐδὲ γινώσκει αὐτό ὑμεῖς δὲ γινώσκετε αὐτὸ, ὅτι παρ᾽ ὑμῖν μένει καὶ ἐν ὑμῖν ἔσται
The Greek word ἄλλον ("another") implies that there is an original Comforter. As mentioned, the original Comforter is Yeshu'a the Anointed One. The one who is "another Comforter" is the Holy Spirit.
In addition, regarding the Spirit of truth, Yeshu'a mentioned that "he dwells with you, and he shall be in you." The Greek verb μένει, translated as "dwells," is in the present tense, meaning that the Spirit of truth was dwelling with the apostles, right then, and right there, in the present. Muḥammad was not born for another 500-600 years. Therefore, Muḥammad could not be the other Comforter, the Spirit of truth. In addition, even Muslims would never admit that Muḥammad is actually in them. On the other hand, the Holy Spirit dwells in Christians (cp. 2 Tim. 1:14).
The reason that the Spirit of truth was dwelling with the apostles in the present, although the Holy Spirit would not be given until Pentecost (cp. Acts 2:1), is because the Holy Spirit is also the Spirit of the Son, Yeshu'a the Anointed One (cp. Rom. 8:9). Yeshu'a was dwelling with the apostles, and there exists a divine community between the Father, the Son, and the Holy Spirit --- the Holy Trinity.
The Prophecy of Yo'el
Many Muslims assert that the Holy Spirit cannot be the "Spirit of truth" or the "Comforter" because the Holy Spirit had always been with the faithful. However, there is no evidence that everyone who had faith in God prior to Pentecost possessed the Holy Spirit. Rather, God gave the Holy Spirit to whom He willed.
The prophet Yo'el prophesied of a time when God would pour out His Spirit (i.e., the Holy Spirit) upon all flesh, meaning both Jews and Gentiles, bondmen and freemen, men and women, children and adults, rather than upon select individuals. Logically, if God was going to pour out His Spirit upon all flesh, then it had not been done so prior to that time.
In Joel 2:28-29, it is written,
28 And it shall come to pass afterwards, I will pour out My Spirit upon all flesh, and your sons and your daughters shall prophesy, your old men shall dream dreams, your young men shall see visions. 29 And in those days, I shall also pour out My Spirit upon the servants and upon the handmaids.
וְהָיָה אַחֲרֵי־כֵן אֶשְׁפֹּוךְ אֶת־רוּחִי עַל־כָּל־בָּשָׂר וְנִבְּאוּ בְּנֵיכֶם וּבְנֹותֵיכֶם זִקְנֵיכֶם חֲלֹמֹות יַחֲלֹמוּן בַּחוּרֵיכֶם חֶזְיֹנֹות יִרְאוּ וְגַם עַל־הָעֲבָדִים וְעַל־הַשְּׁפָחֹות בַּיָּמִים הָהֵמָּה אֶשְׁפֹּוךְ אֶת־רוּחִי׃
In the New Testament, the apostle Petros confirms that this event occurred at Pentecost.
And when the day of Pentecost was fully come, they were all of one accord in one place.
And they were all filled with the Holy Spirit, and they began to speak with other tongues, as the Spirit gave them utterance.
But this is that which was spoken by the prophet Yo'el, "And it shall come to pass in the last days," said God, "I will pour out of My Spirit upon all flesh, and your sons and your daughters shall prophesy, and your young men shall see visions, and your old men shall dream dreams. And in those days, I shall pour out My Spirit upon My servants and upon My handmaidens, and they shall prophesy.
A Contrary Gospel
There is one more thought that I feel is necessary to include. In his epistle to the Galatians 1:8-9, the apostle Paulos wrote,
8 But if we, or an angel from heaven, preach [a gospel] to you other than what we have preached to you, let him be anathema! 9 As we said before, so now I say again --- if anyone preaches [a gospel] other than what you have received, let him be anathema!
8 ἀλλὰ καὶ ἐὰν ἡμεῖς ἢ ἄγγελος ἐξ οὐρανοῦ εὐαγγελίζηται ὑμῖν παρ᾽ ὃ εὐηγγελισάμεθα ὑμῖν ἀνάθεμα ἔστω 9 ὡς προειρήκαμεν καὶ ἄρτι πάλιν λέγω εἴ τις ὑμᾶς εὐαγγελίζεται παρ᾽ ὃ παρελάβετε ἀνάθεμα ἔστω
Since the revelation of Muḥammad which he claimed to have received from an angel contradicts the Gospel preached by Paulos and the other apostles, Muḥammad is considered anathema, or "accursed."