theonewhopostskoreanmen-deactiv:

Cyworld Digital Music Awards:

• 2007: Best Rookie Group
• 2009: Song of the Month (January) and Bonsang Award

Golden Disk Awards:

• 2007: Popularity Award as well as the Rookie Award.
• 2009: Digital Daesang Award and Digital Bonsang Award. (Gee) 
• 2010: Popularity Award, Disk Bonsang Award, Disk Daesang Award. (Oh! second album)
• 2011: Digital Bonsang Award and Digital Daesang Award. (The Boys)

Melon Music Awards:

• 2009: 2009 Top 10, 2009 Artist of the Year, 2009 Smart Radio, 2009 Mobile Music, 2009 Song of the Year (Gee), 2009 Odyssey (Gee).
• 2010: 2010 Top 10, 2010 Artist of the Year, 2010 Best Dressed Singer, 2010 Hot Trend.

Seoul Music Awards:

• 2008: Best Newcomer Award, High1 Music Award, and Popularity Award.
• 2010: Bonsang Award, Daesang Award, Digital Music Award, and Popularity Award. (Gee)
• 2011: Bonsang Award, Daesang Award, Hallyu Special Award, and Popularity Award. (Oh!)
• 2012: Popularity Award and Bonsang Award. (The Boys)

Annual Korean Entertainment Arts Awards:

• 2007: Best New Female Group
• 2008: Best Female Group
• 2010: Best Female Group

Korean Music Awards:

• 2010: Song of the Year (Gee) and Group Musician of the Year Netizen Vote

Mnet Asian Music Awards (MAMA):

• 2010: Best Female Group
• 2011: Artist of the Year

Mnet 20’s Choice Awards:

• 2008: Hot Sweet Music Award
• 2012: Hot Trendy Music Award

Gaon Chart Awards: 

• 2012: Album of the Year Award and Oricon Hallyu Singer Award
• 2013: Album of the Year Award and Song of the Year

KBS Music Festival:

• 2010: Song of the Year (Oh!)

Asia Song Festival:

• 2008: The Best Asian Artist Award
• 2009: Asian Best Group Award
• 2011: Asian Best Group Award

Yahoo! Asian Buzz Award:

• 2009: Top Buzz Star: Female Singer Category
• 2010: Top Buzz Music Video (Oh!)

SBS MTV Best of the Best:

• 2011: Best Girl Group

Nickelodeon Korea Kids’ Choice Awards:

• 2009: Favorite Female Singer
• 2010: Favorite Female Singer
• 2013: Favorite Female Singer

And that was just Korean-wise. Here’s Japanese-wise:

Japanese Gold Disc Awards:

• 2011: The Best 5 New Artists (Domestic) and New Artist of the Year Award
• 2012: Album of the Year (Asia) and Best 3 Albums (Asia)
• 2013: Song of the Year by Download (Asia, Paparazzi) and Best Music Video (Asia, The 1st Japan Tour DVD)

Japan Record Awards:

• 2010: New Artist Award

Space Shower Music Video Awards:

• 2011: Best Pop Music Video Award (Genie)

MTV Music Video Awards Japan

• 2011: Best Group Video (Genie) and Best Karaoke! Song (Genie)
• 2012: Album of the Year (Girls’ Generation)

International award-wise:

Soompi Gayo Awards:

• 2010: Artist of the Year Award

World Music Awards: (Pending)

VEVOCertified Awards:

• 2013: Gee with 100,000,000 views

YouTube Music Awards:

• 2013: Video of the Year Award (I Got A Boy)

Also there are other awards they have won such as:

1. Young Artist Award
2. Artist Daesang 
3. Teen Musical Artist Award
4. Best Picture for Variety
5. Ahjumma Group Award
6. Special Award (Best Singer)
7. Content Industry Award (Special Honor)
8. Singer Award
9. Best Singer Award
10. Best Advertising Models Award
11. Best Singer Dresser
12. Pop Culture Award
13. Art Category
14. Best Single
15. Album of the Year
16. Best Korean Artist
17. Music Video of the Year
18. Asia Star Award
19. Pop Music Award
20. Bonsang Award 
21. Prime Minister’s Award
22. Best New Artist
23. Hallyu Grand Award
24. Special Award for Female Category: Singer
25. The Most Popular Asian Artist
26. Best Jeanist
27. Best Global Icon Group

Oh, let’s not forget each award earned by individual members of the group:

• Taeyeon: Yepp Popularity Award
• Yoona: Best New Actress, Netizen Award 208, Netizen Award 2012
• Seohyun: Most Popular Couple (We Got Married)
• Yuri and Tiffany: Special Award: MC Division
• Taeyeon: Hot DJ, Best OST
• Yoona: Best New Actress and Popularity Drama Actress
• Yuri: Popularity Drama Actress
• Yoona: Netizen Award
• Jessica: Popularity Award 
• Jessica: Korean Barbie 
• Taeyeon: Female Top Buzz Star, Female Radio Award, Radio Newcomer Award
• Seohyun: Recognition Award
• Taeyeon: Best Song From a Series, Song of the Month: February
• Yuri: New Star Award
• Sooyoung: Best K-Style Award

So yep. You’re totally correct. They’ve only won one award.