Yes, it certainly does. Aside from the many biblical and extra-biblical quotations and allusions throughout the Book of Mormon, there are also very specific references to the Bible as a holy text:

2 Nephi 29:3–6

The Gentiles shall say: A Bible! A Bible! We have got a Bible, and there cannot be any more Bible.

But thus saith the Lord God: O fools, they shall have a Bible; and it shall proceed forth from the Jews, mine ancient covenant people. And what thank they the Jews for the Bible which they receive from them? Yea, what do the Gentiles mean? Do they remember the travails, and the labors, and the pains of the Jews, and their diligence unto me, in bringing forth salvation unto the Gentiles?

O ye Gentiles, have ye remembered the Jews, mine ancient covenant people? Nay; but ye have cursed them, and have hated them, and have not sought to recover them. But behold, I will return all these things upon your own heads; for I the Lord have not forgotten my people.

Thou fool, that shall say: A Bible, we have got a Bible, and we need no more Bible. Have ye obtained a Bible save it were by the Jews?

The KJV notion of a “Bible” comes from Classical Greek biblion “book” (byblion, byblios), and biblical Greek biblos “scroll, book,” which translates Hebrew ספר sēfer (Exodus 17:14, 32:32). Greek biblos is derived from the toponym Byblos, the Syrian coastal port-city renowned for the export of “paper,” since Greek biblos originally referred to “Egyptian papyrus; paper” (“leaves of byblus” Herodotus); “book (paper),” usually as biblos “book,” plural bybla “anthology.”[1] Or bibliothēkē “library” (Hebrew ספר sēfer Esther 2:23; bēyt-sifrayyā’ Ezra 6:1; 2 Maccabees 2:13), with biblical Aramaic sěfar “book.”

This leads to terms such as sēfer habberît “Book of the Covenant; Pentateuch” (Exodus 24:7, 2 Kings 23:2) = biblos diathēkēs “book of the Covenant” (Ben Sirach/Ecclesiasticus 24:23). Likewise, the 24 books of the Hebrew Canon/Tanakh (2 Esdras 14:44-45), ta biblia ta hagia “the holy books of Scripture” (1 Maccabbees 12:9); pente bibliōn . . . enos syntagmatos epitemein “five volumes/books . . . condense/abridge into a single book” (2 Maccabees 2:23); tēn hieran biblon “holy book” (2 Maccabees 8:23); biblos “book” (Ben Sirach/ Ecclesiasticus 50:27, 1 Maccabees 3:48; Matthew 1:1, 12:26, Luke 20:42, Acts 1:20, Revelation 20:12,15, 22:7-19) < Hebrew ספר sēfer “book” (Genesis 5:1, Exodus 17:14, Numbers 21:14, Deuteronomy 31:26, Joshua 1:8, 1 Samuel 10:25, 2 Samuel 1:18, 2 Kings 22:8, 23:21,28, 2 Chronicles 35:12,27, Psalm 56:8, Isaiah 29:11-12, Jeremiah 36:2-32, Nahum 1:1, Malachi 3:16, Ezra 6:18, Nehemiah 8:1, 9:3, 12:23, 13:1).

Other such terms include Hebrew מקרא miqrā’ “reading; Holy Writ” (Nehemiah 8:8); Hebrew כתב kātāb, hakkātûb “That which is written; Scripture” (Ezekiel 13:9, Daniel 10:21) = Greek pasa graphē (John 2:22, Acts 8:32, 2 Timothy 3:16), “the scriptures” (Mark 12:24, 1 Corinthians 15:3-4).

[1] Edward Lipiński, Semitic Languages: Outline of a Comparative Grammar, 2nd ed., OLA 80 (Leuven: Peeters, 2001), § 65.8, Assyro-Babylonian niāru, nayāru “papyrus,” from Late Egyptian & Demotic n-yr < ny itrw “belonging to the Nile” > Mishnaic Hebrew nĕyar “paper.”

See also Yanick Rochon, “Does the Book of Mormon ever talk about the Bible?” Quora, May 24, 2019, online at https://qr.ae/TWGGNJ .