I've heard a few times that there is some verse in the bible that instructs people to be hesitant to quote scripture when wanting to "help" those that are hurt, guilty of an offense, looking just to be heard, etc. Clearly, Job's friends are an example of why sometimes "helping" can be hurtful.... But is there explicit text that discourages quoting scripture or "righteously correcting" people when the time is inappropriate? [more explicit than Ecclesiastes 3, for example]
First, consider your own integrity and maturity and decide if the words you want to speak are ones you should listen to yourself. Matthew 7 says:
“Why do you look at the speck of sawdust in your brother’s eye and pay no attention to the plank in your own eye? 4 How can you say to your brother, ‘Let me take the speck out of your eye,’ when all the time there is a plank in your own eye? 5 You hypocrite, first take the plank out of your own eye, and then you will see clearly to remove the speck from your brother’s eye.
Second, check your attitude. The key phrase is "speaking the truth in love". Ephesians 4 says:
As a prisoner for the Lord, then, I urge you to live a life worthy of the calling you have received. 2 Be completely humble and gentle; be patient, bearing with one another in love. 3 Make every effort to keep the unity of the Spirit through the bond of peace. 4 There is one body and one Spirit, just as you were called to one hope when you were called; 5 one Lord, one faith, one baptism; 6 one God and Father of all, who is over all and through all and in all.
7 But to each one of us grace has been given as Christ apportioned it. 8 This is why it says:
“When he ascended on high, he took many captives and gave gifts to his people.”[b] 9 (What does “he ascended” mean except that he also descended to the lower, earthly regions? 10 He who descended is the very one who ascended higher than all the heavens, in order to fill the whole universe.) 11 So Christ himself gave the apostles, the prophets, the evangelists, the pastors and teachers, 12 to equip his people for works of service, so that the body of Christ may be built up 13 until we all reach unity in the faith and in the knowledge of the Son of God and become mature, attaining to the whole measure of the fullness of Christ.
14 Then we will no longer be infants, tossed back and forth by the waves, and blown here and there by every wind of teaching and by the cunning and craftiness of people in their deceitful scheming. 15 Instead, speaking the truth in love, we will grow to become in every respect the mature body of him who is the head, that is, Christ. 16 From him the whole body, joined and held together by every supporting ligament, grows and builds itself up in love, as each part does its work.
Third, become wise and well-informed through careful study. 2 Peter 3 says:
Bear in mind that our Lord’s patience means salvation, just as our dear brother Paul also wrote you with the wisdom that God gave him. 16 He writes the same way in all his letters, speaking in them of these matters. His letters contain some things that are hard to understand, which ignorant and unstable people distort, as they do the other Scriptures, to their own destruction.
Fourth, decide based on your knowledge of the other person whether there is any chance that they will respond favorably, or if you are needlessly risking harm to yourself. Matthew 7:6 says:
6 “Do not give dogs what is sacred; do not throw your pearls to pigs. If you do, they may trample them under their feet, and turn and tear you to pieces.
Fifth, do not despair of being helpful. James 5 says,
19 My brothers and sisters, if one of you should wander from the truth and someone should bring that person back, 20 remember this: Whoever turns a sinner from the error of their way will save them from death and cover over a multitude of sins.
While not as focused as the OP's question demands, I did come across this Proverb:
26:7 Like the useless legs of one who is lame is a proverb in the mouth of a fool.
26:9 Like a thornbush in a drunkard’s hand is a proverb in the mouth of a fool.
If silence were required, that is probably a good description of someone who can't shut up.
One verse that captures the idea of appropriate timing is Ephesians 4:29:
Let no corrupting talk come out of your mouths, but only such as is good for building up, as fits the occasion, that it may give grace to those who hear. [ESV]
Similarly, Matthew 18:15–17 describes the process for addressing sin in a fellow Christian—it should be addressed privately, and only escalated if he does not listen:
15 “If your brother sins against you, go and tell him his fault, between you and him alone. If he listens to you, you have gained your brother. 16 But if he does not listen, take one or two others along with you, that every charge may be established by the evidence of two or three witnesses. 17 If he refuses to listen to them, tell it to the church. And if he refuses to listen even to the church, let him be to you as a Gentile and a tax collector.
Neither passage explicitly mentions "quoting scripture" to someone, but both communicate the importance of choosing the right circumstances to exhort a fellow believer.