Looking at the senses of the verb πιστεύω (believe) and the noun πίστις (faith) in the New Testament graphed at the bottom, the meanings are basically the same. Most commentators did not see a significance in John not using the noun. Bulm is the only commentator I found who commented on this. Blum pointed out that most of the occurrences of πιστεύω are in the present tense; thus emphasizing "active, continuous, and vital trust in Jesus." John emphasized the action of the verb over the noun.
The key word in the Gospel of John is “believe” (pisteuō), which occurs 98 times. The Greek noun “faith” (pistis) does not occur. (A few times, however, the NIV translates the Gr. verb with the Eng. “put … faith in.”) The Greek verb pisteuō is frequently used in the present tense and in participial forms. Apparently John wanted to stress an active, continuous, and vital trust in Jesus.
Blum, E. A. (1985). John. In J. F. Walvoord & R. B. Zuck (Eds.), The Bible Knowledge Commentary: An Exposition of the Scriptures (Vol. 2, p. 270). Wheaton, IL: Victor Books.
Senses of the verb in the New Testament (from Logos Bible Software)
Senses of the noun in the New Testament
A side issue is the Hebrew word the verb translates in the Septuagint (LXX). This is the root of אָמֵן .. adv. verily, truly (Brown, F., Driver, S. R., & Briggs, C. A. (1977). Enhanced Brown-Driver-Briggs Hebrew and English Lexicon (p. 53). Oxford: Clarendon Press.). This Hebrew word is our word amen ἀμήν. Jesus used this Hebrew adverb 50 times in 25 verses of John.
While this example is from Luke 16:11, it gives insight into the language Jesus used in the discourses in the Gospel of John. Note how the words based on this root are common in the Gospel of John.
Second, Jesus asks the pointed question, “If you have not been faithful in unrighteous mammon, who will entrust the truth to you?” (Lk 16:11, my translation). This text exhibits a play on words in Aramaic, which was the language Jesus spoke at home. He says:
If you have not been amin [faithful]
in the unrighteous mammon [your material possessions]
the amuna [the truth]
who will ja’min ith kun [entrust to you].
The root amn, which appears in the word amen, is used here four times. It makes the point that anyone who cheats on his or her taxes will never understand the gospel. Those who have been unfaithful before God with material possessions cannot expect God to reveal his greater treasure to them, which is the truth of God. -- Bailey, K. E. (2008). Jesus through Middle Eastern Eyes: Cultural Studies in the Gospels (pp. 378–380). Downers Grove, IL: IVP Academic.
If then you have not been faithful in the unrighteous wealth, who will entrust to you the true riches? (ESV)
εἰ οὖν ἐν τῷ ἀδίκῳ μαμωνᾷ πιστοὶ οὐκ ἐγένεσθε,* τὸ ἀληθινὸν τίς ὑμῖν πιστεύσει; (NA28)