Most scholars accept the Gospel begins by referencing the opening words of the Old Testament, in the beginning (ἐν ἀρχῇ)... a phrase also present in Proverbs (see discussion below). Yet a comparison of key words shows there no significant allusions to the texts of Proverbs:

None of the words in John were used in LXX-Proverbs. In addition to distinguishing God as Lord, Proverbs describes wisdom's relationship with the Lord using different language and never states directly or indirectly that wisdom actually was God (or a god). Finally, when the Gospel writer expresses a similar idea like "with God", they avoided the language of Proverbs.
In addition, it is apparent there is little textual correspondence between the Greek translations of Genesis and Proverbs, and where Proverbs does have similar concepts as that of Genesis, the original wisdom writer chose to express them with different words (i.e. God, אלהים/יהוה, made or create, קנה/ברא, work, מפעל/מלאכת) something the LXX translation occasionally misstates.
The Word was with God
One of the most important concepts in the Prologue is the Word was with God. The Greek is πρὸς which "expresses direction...'in the direction of'...marker of movement or orientation toward someone/something..."1
Some say the phrase could be rendered "...the Word was turned toward God..."2
The concept continues throughout in the Gospel: the Word "with" God was sent, and returned. "Return" then is an actual re-"turn"-ing back to and so will see God (John 1:18).
Neither ἦν nor πρὸς are used in the Greek translation of Proverbs:
22 The Lord created me as the beginning of his ways, for the sake of his works. 23 Before he made the present age he founded me, in the beginning. 24 Before he made the earth and before he made the depths, before he brought forth the springs of the waters, 25 before the mountains were established and before all the hills, he begets me. 26 The Lord made countries and uninhabited spaces and the habitable heights of what is under heaven. 27 When he prepared the sky, I was present with him, and when he marked out his own throne on the winds. 28 When he made strong the clouds above and when he made secure the springs of what is under heaven, 29 when he made strong the foundations of the earth, 30 I was beside him, fitting together; it is I who was the one in who he took delight. And each day I was glad in his presence at every moment, 31 when he rejoiced after he had completed the world and rejoiced among the sons of men.
(LXX-Proverbs NETS)
22 κύριος ἔκτισέν με ἀρχὴν ὁδῶν αὐτοῦ εἰς ἔργα αὐτοῦ 23 πρὸ τοῦ αἰῶνος ἐθεμελίωσέν με ἐν ἀρχῇ 24 πρὸ τοῦ τὴν γῆν ποιῆσαι καὶ πρὸ τοῦ τὰς ἀβύσσους ποιῆσαι πρὸ τοῦ προελθεῗν τὰς πηγὰς τῶν ὑδάτων 25 πρὸ τοῦ ὄρη ἑδρασθῆναι πρὸ δὲ πάντων βουνῶν γεννᾷ με 26 κύριος ἐποίησεν χώρας καὶ ἀοικήτους καὶ ἄκρα οἰκούμενα τῆς ὑπ᾽ οὐρανόν 27 ἡνίκα ἡτοίμαζεν τὸν οὐρανόν συμπαρήμην αὐτῷ καὶ ὅτε ἀφώριζεν τὸν ἑαυτοῦ θρόνον ἐπ᾽ ἀνέμων 28 ἡνίκα ἰσχυρὰ ἐποίει τὰ ἄνω νέφη καὶ ὡς ἀσφαλεῗς ἐτίθει πηγὰς τῆς ὑπ᾽ οὐρανὸν 29 καὶ ἰσχυρὰ ἐποίει τὰ θεμέλια τῆς γῆς 30 ἤμην παρ᾽ αὐτῷ ἁρμόζουσα ἐγὼ ἤμην ᾗ προσέχαιρεν καθ᾽ ἡμέραν δὲ εὐφραινόμην ἐν προσώπῳ αὐτοῦ ἐν παντὶ καιρῷ 31 ὅτε εὐφραίνετο τὴν οἰκουμένην συντελέσας καὶ ἐνευφραίνετο ἐν υἱοῗς ἀνθρώπων
The absence of this essential statement is direct proof against an allusion to Proverbs. When wisdom states she was "with" the Lord (8:27), the word used to describe that condition is συμπαρήμην, which is "to be present" as in (Acts 25:24). It would have been a simple matter for the Gospel writer to say "...the Word was present with God..." to establish a clear connection to Proverbs 8:27. In fact, "to be present" is how most understand the phrase "the Word was with God." Clearly the Gospel writer avoided making this linguistic connection.
Later wisdom states she was "by" him. Here the word is παρά which is used in the Prologue (1:6, 14). However, the Gospel use conveys the opposite meaning. John the Baptist was not in proximity to God: he was sent "by" God. Likewise, when the Word is revealed as becoming flesh, it was on earth with those who beheld Him. Again, it would have been as simple matter for the Gospel writer to use the words as they were in Proverbs, but as with πρὸς, they chose a message distinct from, and in this case, opposite to the use with wisdom in Proverbs.
Additional evidence is seen in the etymology of πρὸς which is the strengthened form of πρὸ. πρὸ is used 6 times in Proverbs (8:23, 24 [3 times], 25 [2 times]) and 9 times in John (1:47; 5:7; 10:8; 11:55; 12:1; 13:1; 13:19; 17:5; 17:24). The Gospel writer employed πρὸ in a manner which alludes to "with you before..." and "before the foundation..." as found in Proverbs:
And now, Father, glorify me in your own presence with the glory that I had with you before the world existed. (John 17:5) [ESV]
καὶ νῦν δόξασόν με σύ πάτερ παρὰ σεαυτῷ τῇ δόξῃ ᾗ εἶχον πρὸ τοῦ τὸν κόσμον εἶναι παρὰ σοί
Father, I desire that they also, whom you have given me, may be with me where I am, to see my glory that you have given me because you loved me before the foundation of the world. (John 17:24)
πάτερ ὃ δέδωκάς μοι θέλω ἵνα ὅπου εἰμὶ ἐγὼ κἀκεῖνοι ὦσιν μετ᾽ ἐμοῦ ἵνα θεωρῶσιν τὴν δόξαν τὴν ἐμὴν ἣν δέδωκάς μοι ὅτι ἠγάπησάς με πρὸ καταβολῆς κόσμου
As with πρὸς, the Gospel writer avoids making a direct connection, and they significantly alter the message from that found in Proverbs. In 17:5, it is "glory" not "wisdom" which was with the Father and in 17:24, it is "love." Moreover, the Gospel writer replaced "foundation of the earth" with "foundation of the world," κόσμος (a word not used in the passage in Proverbs). Effectively, the use is "allusion avoidance" as if the Gospel writer was aware of the potential to connect the Word to the personified wisdom of Proverbs, and purposely wrote to avoid making any connection. Obviously, it would have been a simple matter to write "foundation of the earth." In fact, like "with God," most understand "foundation of the world" to be equivalent with "foundation of the earth" despite the obvious difference.
Make, Create, or Possess
There is a interesting question on the translation of κτίζω which is "create" in Proverbs. Regardless of how the word was understood, it is not used in the Gospel. That is, there is no allusion to "create" between the Gospel and Proverbs.
The Prologue only employs the verb γίνομαι to describe a work of creation (John 1:3, 10, 12, 14). As with πρὸς, γίνομαι is not used in Proverbs. It is however found in the LXX Genesis account of creation (1:3, 5, 6, 8, 9, 11, 13, 14, 15, 19, 20, 23, 24, 30, 31).
A second issue with trying to connect creation as described in John with that in Proverbs, is the manner in which how the work of making things is conveyed. In John, the Word has an active role; in Proverbs, wisdom is a witness to what is made and nothing is made through her. On the other hand, the Word is actively participating in what is made: all things are made through him and nothing which is made (i.e. lady wisdom) is made without Him. In the Gospel through is διά, and as with πρὸς and γίνομαι, διά is not used in the passage in Proverbs.
In the beginning
The one common phrase to all three is ἐν ἀρχῇ: in the beginning...
Genesis: In the beginning God ἐποίησεν the heaven and the earth
Proverbs: The Lord ἔκτισέν me the beginning of his ways for his works
Proverbs: He ἐθεμελίωσέν me before time in the beginning, before he made the earth
John: In the beginning ἦν the Word
Before looking at these in detail, it is important to acknowledge John unequivocally states nothing came into existence except through the Word (1:3). So it is impossible for the Word, as understood by John, to be "created" by some other agency. Rather, according to the Gospel writer, if wisdom was created, it was through the Word. Also, "create" in Proverbs is קָנָה which means possessed. While wisdom can be possessed, it makes little sense to say it was created (God must have wisdom). Similarly when the OP refers to "the Word" as the utterance of God, an "utterance" from God is not created. These are always present with God.
There are two statements in Proverbs: both fail to follow Genesis. The first, 8:22 speaks to קָנָנִי רֵאשִׁית "possessing (or creating) in the beginning," not בְּרֵאשִׁית בָּרָא "in the beginning made" as in Genesis 1:1. The second 8:23 speaks of נִסַּכְתִּי מֵרֹאשׁ "I was set up from the beginning" using words not found in the Genesis account of creation. Therefore, in the original text, there is no clear allusion to creation as detailed in Genesis.
In LXX-Proverbs ἀρχὴν is in v. 22 and ἐν ἀρχῇ in v 23. Thus, ἐν ἀρχῇ is properly speaking to נִסַּכְתִּי מֵרֹאשׁ "I was set up from the beginning...before the earth was made." Verse 22 which speaks to being "possessed" uses ἀρχὴν to mean "the head or the beginning" of His ways. In other words, the reference is not to time. It is verse 23 which has the element of time with "established...before the earth was made." Notably absent from this ἐν ἀρχῇ passage is "heaven." That is, the Genesis account speaks to heaven and earth where Proverbs has only "earth." The passage in Proverbs may be linked theoretically to "creation" in Genesis, but connecting the literal texts is tenuous at best, and all allusions in John 1 are to clearly to Genesis.
Conclusion
Despite English translations which might suggest John has something in common with Proverbs 8:22-31, there is no allusion in the Greek texts to the main theme the Word was with God; how things were made and the significance of being with the Father from the foundation of the world.
Notes:
1. Fredrick William Danker, A Greek-English Lexicon of the New Testament and Other Early Christian Literature, The University Chicago Press, 2000, p. 873-874. [Eg. Proverbs 9:4: “He who is a fool, let him turn aside to me…” ὅς ἐστιν ἄφρων ἐκκλινάτω πρός με]
2. Francis J. Moloney, Beginning the Good News: A Narrative Approach, Wipf & Stock Publishers, 1992, p. 138