The question should be put more precisely and with a greater clarity, for it is unclear whether it asks:
a) Do the New Testament writers consider in their account of Jesus' letting His disciples to address God as their Father as a change in relationship of humans with God towards a greater intimacy?
b) Is that, so to say, "Father" appellation introduced by the New Testament authors really/objectively/ontologically/historically/psychologically/morally, notwithstanding the intentions or agenda of the NT writers, imply any essential promotional change with respect of human intimacy with God, as compared to the texts of the Old Testament, which also convey a story of the human-God relationship with a language of perhaps not a lesser intimacy, which is to be discussed?
If the "a)" is asked, then, yes, of course, 100%! - For the New Testament authors clearly assert that in relationship of mankind and God something radically changed and humans are already invested with a totally novel and hitherto unprecedented in history authority of becoming children of God (John 1:12) and co-heirs of Jesus from Jesus Himself, who, as God's only begotten Son has this authority properly and naturally (for the very term "Son" means that He possesses the entire inheritance and authority of God-His-Begetter.) For men do not say boldly "Father" to God out of self-given, self-invested authority, but through the Holy Spirit that Jesus asked Father to send them (John 14:16), and exactly through this Holy Spirit, called by Paul also the "Spirit of Sonship", humans can authoritatively cry to God in the most intimate manner possible - "Abba", which means, "Father" (Rom.8:15). Moreover, this status is incomparably, unimaginably and unutterably higher than not only that of all the prophets of the Old Testament, but even that of the pre-lapsarian Adam and Eve in paradise, for they could not enjoy even being in paradise that glory that God gave through His Son to humans after the latter's Incarnation, for the Kingdom of Heaven, to which humans have already access through Christ is higher than Adam's paradise beyond comparison.
Thus, God's eternal Logos-incarnate, Jesus Christ, vouchsafed to humans to be co-heirs of Himself, which is the impenetrable and awesome mystery of us, the created beings, becoming co-heirs of Him, the Creator, whose Being is eternal and co-eternal with the Father and the Spirit of Sonship that goes out from the Father (John 15:26).
But if the "b)" is asked, then totally different discussion will ensue, in which I will abstain at a moment indulging myself - although I fully believe that the NT authors convey not only their intentional and subjective, but also real/ontological/objective truth - but to sum up my post: the question should be presented with a greater clarity, otherwise the answers will be as blurred as the question itself.