For Psalms 14 & 53 as a whole, mostly "yes"
A few opening thoughts:
Yes, it does seem like he is using or borrowing from the Septuagint in particular.
Yes, it seems to be a kind of quote/reference. But, New Testament authors did not "quote" the Old Testament as we mean "quote" today; they would use a "rephrased rerefence" (explained later in the "Considerations" section). That does seem to be the case here, though.
This seems to be more a reference to Psalms 14 & 53 (and Jewish discussions about them) as a whole than it is to any single verse in particular, though the verses in the OP do certainly apply.
"Which specific Greek translation" Paul was using is probably a question we have thought about more than Paul had because the New Testament contemporaries didn't think that way. In Paul's mind, as in his audience's, he wasn't "quoting the Septuagint" or any other translation; he was "referring to Psalms".
Realistically, Paul isn't writing (dictating) in "Greek", it's more or less "Greekbrew" (Hebrew native, Greek franca lingua; comparable to what we in Taiwan call 'Chinglish' or in America 'Spanglish'). As a member of the Sanhedrin, Paul was more familiar with the Hebrew Psalms than the Septuagint. So, while all the learned Jews got together and talked in this Greco-Roman "Greek language thing", they were still thinking in Hebrew. That being said, next thought...
"None is righteous" vs "All are unrighteous"—and differences like that—surmount to "six" vs "half a dozen". Those kinds of differences will be normal when speaking and dictating correspondence in "Greekbrew" like Paul is doing. In a casual conversation about this, Paul and other "Greekbrew" speakers like Paul and Peter and the crew might not even remember if they said "none does right" or "everyone is unfair" anymore than a native English speaker remembers pronouncing "the" as [thu] or [thee] when trying to speak more clearly over the phone; both ways of speaking seem the same to native English speakers, Paul just the same. There are too many language factors to determine whether Paul was trying to make the audience think about the Septuagint in particular as much as he was referring to Jewish discussions (which Greeks, also, were learning about in the early Church) surrounding said Psalms.
The biggest argument for the Septuagint is that it was most-widely used. Did the Septuagint almost surely influence Paul's "Greebrew"—absolutely! There is both cultural-historical-contextual evidence (the Septuagint was widely used) and there is textual...
With those considerations in mind, let's look at the actual texts...
Comparison of the passages in Greek
Italics text shows same words in Romans 3:10 and Psalms
Romans 3:10b — Footnote 1
Οὐκ ἔστιν δίκαιος οὐδὲ εἷς,
Lit def: not is righteous not-even one
Psalm 53:1c — Footnote 2
οὐκ ἔστιν ποιῶν ἀγαθόν
Lit def: not is do good
Psalm 14:3 has some problems in the Septuagint because the verse is incredibly long, as discussed and translated here.
Psalm 14:3 — Footnote 3
πάντες ἐξέκλιναν ἅμα ἠχρεώθησαν οὐκ ἔστιν ποιῶν χρηστότητα οὐκ ἔστιν ἕως ἑνός τάφος ἀνεῳγμένος ὁ λάρυγξ αὐτῶν ταῖς γλώσσαις αὐτῶν ἐδολιοῦσαν ἰὸς ἀσπίδων ὑπὸ τὰ χείλη αὐτῶν ὧν τὸ στόμα ἀρᾶς καὶ πικρίας γέμει ὀξεῖς οἱ πόδες αὐτῶν ἐκχέαι αἷμα σύντριμμα καὶ ταλαιπωρία ἐν ταῖς ὁδοῖς αὐτῶν καὶ ὁδὸν εἰρήνης οὐκ ἔγνωσαν οὐκ ἔστιν φόβος θεοῦ ἀπέναντι τῶν ὀφθαλμῶν αὐτῶν
It's the frequent use of "οὐκ ἔστιν" that might make this just as memorable to the NT Jewish audience as saying "I took the road less traveled" would remind English speakers of Robert Frost, being considered a citation in itself even without the author's name.
Psalm 14:3 has as much similarity to the overlap of Romans 3:10 as the small part of Psalm 53:1. Ch14's reference is very frequent and in v3 (not the very first verse), while ch15 is in v1. That balance probably means he is referencing both Psalms about equally.
The identical (and in Psalm 14 frequent) use of "οὐκ ἔστιν", though a common phrase, suggests Paul could be borrowing from the Septuagint because this would certainly appeal to a Septuagint-reading audience. These notes highlight examples of the use with Psalm 53, also with Psalm 14 (13 LXX). Looking at the striking similarities in those early verses in Psalm 14, Paul is probably also referring to discussions about (see opening thoughts) all of Psalm 14 just as all of Psalm 53, not any one verse in particular.
NT authors wouldn't reference only one single sentence (verse divisions didn't exist yet), but use one phrase or sentence to reference the entire pericope with the full context of the passage. They viewed text more holistically, a healthy thing for any study. Paul doesn't teach this, he presumes it and demonstrates it because it was common practice. We would do well to follow that example, but that opens other topics.
Paul, at least, likely demonstrates familiarity with the Septuagint here because the first verse (viz 53:1) would more likely indicate the larger passage he refers to (see example with Jesus, later).
Conclusion: It is indeed a New Testament -style reference to Psalm 53 as a whole (partially quoting from the first verse to indicate the entire psalm, rephrasing it as Paul's way of quickly explaining his interpretation/application of it). By extension, it would apply to related Psalms, such as, but not limited to, Psalm 14 as a whole.
New Testament authors did not have chapter and verse numbers as we do. But, as Jewish children, they studied Moses and The Prophets (what we call the Old Testament) inside and out. So, the way they referred to passages the Old Testament was to quote a memorable or beginning part of the passage, and even "change" it a little so as to explain how they were applying it.
One of the classic examples is Jesus on the cross:
My God, my God, why have You foresaken me?
Matthew 27:46 (NASB) cf Psalm 22:1
Jesus was not merely lacking for words to express himself; he was lacking strength and oxygen to say those words; he was saying that all of Psalm 22 said it all for him.
And, Paul is doing something similar in Romans 3:10.
Still, using Psalms 14 & 53 to interpret Romans 3 would be almost a necessity in diligent exegesis.
Holmes, M. W. (2011–2013). The Greek New Testament: SBL Edition (Ro 3:10). Lexham Press; Society of Biblical Literature.
Blue Letter Bible Psalm 53:1
Blue Letter Bible Psalm 14:3