I don't see any mention of levivot in Talmudic Hebrew, so I assume it was a "renewed" word in Modern Hebrew, taken from the Bible. The word only appears in the story of Amnon and Tamar, Shmuel II, Chapter 13:
ו וַיִּשְׁכַּב אַמְנוֹן, וַיִּתְחָל; וַיָּבֹא הַמֶּלֶךְ לִרְאוֹתוֹ, וַיֹּאמֶר אַמְנוֹן אֶל-הַמֶּלֶךְ תָּבוֹא-נָא תָּמָר אֲחֹתִי וּתְלַבֵּב לְעֵינַי שְׁתֵּי לְבִבוֹת, וְאֶבְרֶה, מִיָּדָהּ. ז וַיִּשְׁלַח דָּוִד אֶל-תָּמָר, הַבַּיְתָה לֵאמֹר: לְכִי נָא, בֵּית אַמְנוֹן אָחִיךְ, וַעֲשִׂי-לוֹ, הַבִּרְיָה. ח וַתֵּלֶךְ תָּמָר, בֵּית אַמְנוֹן אָחִיהָ--וְהוּא שֹׁכֵב; וַתִּקַּח אֶת-הַבָּצֵק ותלוש (וַתָּלָשׁ) וַתְּלַבֵּב לְעֵינָיו, וַתְּבַשֵּׁל אֶת-הַלְּבִבוֹת. ט וַתִּקַּח אֶת-הַמַּשְׂרֵת וַתִּצֹק לְפָנָיו, וַיְמָאֵן לֶאֱכוֹל; וַיֹּאמֶר אַמְנוֹן, הוֹצִיאוּ כָל-אִישׁ מֵעָלַי, וַיֵּצְאוּ כָל-אִישׁ, מֵעָלָיו. י וַיֹּאמֶר אַמְנוֹן אֶל-תָּמָר, הָבִיאִי הַבִּרְיָה הַחֶדֶר, וְאֶבְרֶה, מִיָּדֵךְ; וַתִּקַּח תָּמָר, אֶת-הַלְּבִבוֹת אֲשֶׁר עָשָׂתָה, וַתָּבֵא לְאַמְנוֹן אָחִיהָ, הֶחָדְרָה.
The JPS translation has:
6: Amnon lay down and pretended to be sick. The king came to see him, and Amnon said to the king, "Let my sister Tamar come and prepare a couple of cakes in front of me, and let her bring them to me."
7: David sent a message to Tamar in the palace, "Please go to the house of your brother Amnon and prepare some food for him."
8: Tamar went to the house of her brother Amnon, who was in bed. She took dough and kneaded it into cakes in front of him and cooked the cakes.
9: She took the pan and set out [the cakes], but Amnon refused to eat and ordered everyone to withdraw. After everyone had withdrawn,
10: Amnon said to Tamar "Bring the food inside and feed me." Tamar took the cakes she had made and brought them to her brother inside.
However, the translation of levivot as cakes doesn't fully capture the meaning of the text. Everett Fox writes in his introduction to his translation of Shmuel, Give Us a King!:
The heart, too, comes into play, particularly in the memorable cycle of stories that recounts Absalom's rebellion against his father David (II Sam. 13-20). The tone is set already in the opening story, the rape of Absalom's sister Tamar. Amnon, the crown prince, pretends to be ill and requests that his half sister make levivot, usually translated as "cakes," for him. The noun occurs four times, and the root appears twice more in verbal form. But as some interpreters have noticed, the homonym (levav) means "heart," and the verbal form of l-v-v occurs in the Song of Songs (4:9, "You have captured-my-heart" [JPS]). So a word connected with seduction in love poetry is appropriate enough in the mouth of the lovesick Amnon, and on this and other grounds (see notes below) we are justified in translating levivot as "heart-shaped-dumplings."(See this Hebrew article for more about the symbolism of levivot and lev לב - heart.)
The word "dumplings" also fits the Targum's translation חליטה chalita, which Jastrow defines as "a paste made of flour stirred in boiling water, dumpling". Although Rashi says that the dough was first placed in boiling water, then fried in oil. (Ralbag says that in laaz [foreign language - I assume French] the term is קישפי"ל. I don't know what that is, but it doesn't seem to be related to quiche, which comes from German.)
This article says that levivot can be compared to Akkadian akal lebbu and Greek kolouri. From here I see that kolouri (sometimes spelled culuri) is a kind of bagel, and bagels are indeed dough boiled in water (and then baked.) So Chanukah has a connection to both doughnuts and bagels.