As there are no "secret codes" in Doubutsu no Mori to obtain a 2x1 table for free from Tom Nook, you instead have to get lucky in one of three ways: lost and found, chores, or Nook's shop.

The following are all 2x1 table names in the Japanese with their English (ACGCN) equivalents:

Kana (Transliteration) - English (cost from Nook)

アジアなテーブル (ajia na teeburu) - Exotic Table (Redd only) アジアなバンダヂ (ajia na bandaji) - Exotic Chest (1800 Bells) カントリーなつくえ (kantorii na tsukue) - Ranch Tea Table (1800 Bells) モノクロローテーブル (monokuro roo teeburu) - Modern Table (1800 Bells) モノクロデスク (monokuro desuku) - Modern Desk (2480 Bells) リゾートなテーブル (rizooto na teeburu) - Cabana Table (1800 Bells) ロイヤルなテーブル (roiyaru na teeburu) - Regal Table (2450 Bells) ログソファテーブル (rogu sofa teeburu) - Cabin Low Table (2100 Bells) じむづくえ (jimu zukue) - Office Desk (1600 Bells) すかしぼりのテーブル (sukashibori no teeburu) - Glass-Top Table (2200 Bells) せんせいのデスク (sensei no desuku) - Teacher's Desk (1580 Bells)

Note not all furniture containing「テーブル」is a 2x1 table (some 2x2 or 1x1), only the above list. The Exotic Table can't be bought from Nook, so it can only be obtained from chores or lost and found.