The Idea in Brief
A survey of early Jewish sources provides insights as to how the Jews understood the meaning of the verb in this verse. Their assessment was that the concubine had gone out to pursue sexually illicit relationships, but only because she felt repugnance toward him because of his behavior toward her. She then left him.
Commentary by Rashi
First, the 12th Century Talmudic scholar, Rabbi Shlomo Yitzchaki (Rashi) made an excellent observation of this verse that helps to understand how the Jews interpreted this verse. Rashi writes the following commentary for this verse:
ותזנה עליו פילגשו: זנתה מביתו אל החוץ כל לשון זנות אינו אלא לשון (תרגום אונקלוס בראשית לד לא): נפקת ברא ,יוצאת מבעלה לאהוב את
And his concubine turned away from him: She turned from his house to the outside. Every expression of [the verb] זָנָה means the expression נפקת ברא, or "[she is one who] goes outside" (per Targum Onkelos Genesis 34:31), [thus the concubine was] departing from her husband to love others.
What Rashi did was look at the Aramaic translation of the Pentateuch, which is the Targum Onkelos that had already appeared in the 2nd Century, and he analyzed the first mention of "harlot[ry]" in the Torah. This targum provided some insight as to how the Jews in the 2nd Century had understood passages in Biblical Hebrew, since translating into Aramaic provided nuances not otherwise evident by just reading Hebrew.
So what Rashi concluded was that Genesis 34 described the accosting of Dinah, the daughter of Jacob, and that the translation into Aramaic provided important information. That is, in Genesis 34:31 two of Dinah's brothers (Levi and Simeon) ask Jacob, "Should he make our sister a harlot?" In this regard, as Rashi notes, the Targum Onkelos translates the phrase into Aramaic as "נפקת ברא," which is a participle that literally means "she [is one who] goes outside [for sexual immorality]." Thus the 2004 edition of the Jewish Publication Society translation of the Hebrew Bible captures Rashi's recommended translation of "And his concubine turned away from him" in the following:
Judges 19:2 (JPS 2004)
2 Once his concubine deserted him, leaving him for her father's house in Bethlehem in Judah; and she stayed there a full four months.
The concubine "deserted him (or went out for sexual immorality), leaving him." The nuance here includes sexual immorality as other oral tradition of the Jews also makes evident.
The Babylonian Talmud
The Babylonian Talmud was codified in the 3rd to 5th Centuries, and claims to have documented the oral tradition of the Jews (since Moses) to include the earliest core commentary (Mishnah) and other later commentary (Gemara) and then supplementary material (Tosefta), which now includes Rashi in modern editions. The Talmud addresses various topics to include Scripture passages and their meaning. The verse in question is in fact discussed in the Talmud. That is, the Jews believed that the verb in this verse suggested that the concubine had gone out and become unfaithful. The Talmud suggests that the Levite was overbearing to the concubine, which precipitated her behavior against him, and thus her repugnance for him. The following excerpt comes from Folio 6B and Folio 7A from Tractate Gittin, which is the part of the Talmud concerning laws on divorce. The following translation comes from Jacob Nuesner.
E. Said to him Abbayye, “So is anybody who doesn’t know what R. Isaac said going to be called unreliable? True enough, if it were a matter of reasoning, well and good. But this is just a matter of knowing a fact of tradition, and he doesn’t happen to have heard that particular tradition! And furthermore, R. Ebiatar is an authority whose ruling was confirmed by his [heavenly] Master, for it is written, ‘And his concubine played the harlot against him’ (Jud. 19:2). R. Ebiatar said, ‘He found a fly on her.’ R. Jonathan said, ‘It was a hair.’ Then R. Ebiatar found Elijah. He said to him, ‘What is the Holy One, blessed be He, working on these days?’ He said to him, ‘He’s occupied with the passage on the concubine in Gibea.’ ‘And what’s he say about it?’ He said to him, ‘My son Ebiatar—this is what he says, and my son Jonathan—this is what he says.’ He said to him, ‘God forbid! Is anything subject to doubt before the Heaven?’ He said to him, ‘Both this position and that represent the words of the living God. He did find a fly, but paid no attention. Then he found a hair, and he paid attention.’ ”
F. Said R. Judah, “It was a fly in a dish, and a hair on ‘that place.’ The fly was merely disgusting, but the hair was dangerous.”
G. There are those who say, “He found both in the dish. The fly was an accident, but the hair was deliberate.”
I.27 A. Said R. Hisda, “A man should never cast too much fear on his household, for lo, as to the concubine of Gibea, he cast too much fear on her, and she caused the death of how many tens of thousands of Israelites.”
The hair on ‘that place’ was her genitalia, as evident in another translation of the same passage from Sefaria.org. Please note that the Talmud accepts the reading of Judges 19:2 which is the same as the Masoretic Text. Also, the Talmud accepts a double entendre of the verb, because Elijah indicates, “...both this position and that represent the words of the living God.” In other words, she went out and committed sexual immorality because of his treatment of her, which left her with a sense of repugnancy for him, which is why she left him. That is, she was not "angry" (per the LXX), but felt a sense of repugnancy toward him.
Targum Jonathan to the Prophets
The last Jewish source is the Targum Jonathan, which first appeared in the Fourth Century. This targum translates the passage at hand into Aramaic.
Judges 19:2 (CAL)
וּבַסַרַת עְלֹוהִי לְחֵינְתֵיה וַאְזַלַת מִלְוָתֵיה לְבֵית אְבוּהָא לְבֵית לַחַם דְבֵית יְהוּדָה וַהְוָת תַמָן יֹומִין אַרבְעָה יַרחִין׃
And his concubine despised him, and she departed his presence to her father's house in Bethlehem in Judah, and was there for a span of four months.
Rashi does not mention this targum, which uses the words "וּבַסַרַת עְלֹוהִי" to indicate that she despised him. These exact same words (with one variant) are found in the same targum in 2 Samuel 6:16, where Michal, the daughter of Saul, despises King David in her heart when she observes him, the new King of Israel, dancing in the street in praise to the Lord. The idea here in both passages is profound contempt for ones husband.
The Jews recognized the ambiguity of this verb in this verse as evident in not only in the Talmud but also two targumim as well. The collective impression of these Jewish sources was that the concubine was unfaithful because the Levite was overbearing with her. She found him repugnant, and so she left him.
Finally, the answers of the two questions of the OP would be as follows:
(1) Does the Hebrew word imply any act of infidelity on the part of the woman or simply that they had a quarrel and she left him, as the Greek has it?
ANSWER: The Hebrew word implies infidelity AND that they had a quarrel AND that she left him.
(2) If she had been a whore or an adulteress would her husband have gone after her to bring her back again?
ANSWER: He went after her because her infidelity stemmed from his treatment of her. In other words, her infidelity was his own fault.