Is there a place in the Bible that says that you shouldn't argue or that it is shameful to God?

Perhaps you are thinking of Job 6:25How forcible are right words! but what doth your arguing reprove?
See All..., which states, "How forcible are right words! but what doth your arguing reprove?"

However, the Bible does not use the word argue much (besides Job 6:25How forcible are right words! but what doth your arguing reprove?
See All..., "arguments" are mentioned in Job 23:4I would order my cause before him, and fill my mouth with arguments.
See All...). A more common word in the Bible is "dispute." All disputations are not bad in the Bible. Paul , when he first got saved (and was still called Saul), "spake boldly in the name of the Lord Jesus, and disputed against the Grecians" (Acts 9:29And he spake boldly in the name of the Lord Jesus, and disputed against the Grecians: but they went about to slay him.
See All...). Later, we find Paul disputing in the synagogues with the Jews (Acts 17:17Therefore disputed he in the synagogue with the Jews, and with the devout persons, and in the market daily with them that met with him.
See All...; 19:8
See All...) and in the school of Tyrannus in the city of Ephesus (Acts 19:9But when divers were hardened, and believed not, but spake evil of that way before the multitude, he departed from them, and separated the disciples, disputing daily in the school of one Tyrannus.
See All...). But these were clearly debates with those who did not know the truth of the gospel about the things of God. It was confrontational witnessing. In another incident, Paul and Barnabus "had no small dissension and disputation" (Acts 15:1-2 [1] And certain men which came down from Judaea taught the brethren, and said, Except ye be circumcised after the manner of Moses, ye cannot be saved. [2] When therefore Paul and Barnabas had no small dissension and disputation with them, they determined that Paul and Barnabas, and certain other of them, should go up to Jerusalem unto the apostles and elders about this question.
See All...) with some who taught the necessity of circumcision for salvation. Therefore, major doctrinal issues may be proper cause for disputation.

However, disputations among believers often take a negative turn. During the ministry of Christ, the disciples "disputed among themselves, who should be the greatest" (Matthew 9:34But the Pharisees said, He casteth out devils through the prince of the devils.
See All...). Jesus rebuked them for their pride. Paul spoke against "doubtful disputations" (Romans 14:1Him that is weak in the faith receive ye, but not to doubtful disputations.
See All...) and commanded the Philippians: "Do all things without murmurings and disputings" (Philippians 2:14For the earth shall be filled with the knowledge of the glory of the LORD, as the waters cover the sea.
See All...). In another place, he warned of "perverse disputings of men of corrupt minds" (1 Timothy 6:5Perverse disputings of men of corrupt minds, and destitute of the truth, supposing that gain is godliness: from such withdraw thyself.
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The believer is commanded: "Be ye angry, and sin not: let not the sun go down upon your wrath" (Ephesians 4:26Be ye angry, and sin not: let not the sun go down upon your wrath:
See All...). Consider this passage:

James 1:19Wherefore, my beloved brethren, let every man be swift to hear, slow to speak, slow to wrath:
See All... Wherefore, my beloved brethren, let every man be swift to hear, slow to speak, slow to wrath:  20 For the wrath of man worketh not the righteousness of God.

Proverbs 16:32He that is slow to anger is better than the mighty; and he that ruleth his spirit than he that taketh a city.
See All... states, "He that is slow to anger is better than the mighty; and he that ruleth his spirit than he that taketh a city." Those who argue often lose control of their spirit and that is to be avoided. So, although I know of no verse stating exactly what you ask, there are many warnings that should make the believer pause and think before he sinks in to a state of argumentation.