The original Baldur’s Gate games does not offer same-sex relationship options. In Baldur’s Gate II: Shadows of Amn, male player-characters can romance Aerie (this romance could be continued into Baldur’s Gate II: Throne of Bhaal) or Jaheira. Also, assuming they are any race but elf they can also romance Viconia. Notably, all of those female NPCs are elves. The only romance option for female player-characters is Anomen (a human cleric), and at least one site indicates that there would have been more but they were cut during the production process. Also, as Vivian Abraham points out, “Anomen is an ass.” She also points out that “there is a disturbing relationship between how evil a female character is and how interested in sex she is in this game” (2012).
That said, many players created mods (or modifications) to add romances to the first two games in the series, many of whom were able to be romanced by male and female player-characters. You can see examples of these here, here, and here. Some were also specifically homosexual mod additions, like Nathaniel. A full list is available here.
In the Enhanced Edition of the games, there are two same-sex romance options. Dorn, a bisexual male half-orc who appears in both games, and Hexxat, a lesbian human vampire who appears in the second game.
In Baldur’s Gate II, the Dorn, a bisexual male half-orc, and Hexxat, a lesbian (much to some players dismay) human vampire, are only available to “evil” players, and thus not available to be romanced by a “good” player-character.

Hexxat
Image credit http://baldursgate.wikia.com/wiki/Hexxat?file=Hexxat_Portrait_BG2.png
You can view a video of how the romance with Hexxat plays out here.

Dorn
Image credit http://baldursgate.wikia.com/wiki/Dorn_Il-Khan?file=DornPortrait.jpg
You can view a video of the romance develop between Dorn and a male player-character here.
Citations:
- Abraham, V. (2012, August 17). Baldur’s Gate Romance. Gaming as Women. http://www.gamingaswomen.com/posts/2012/08/baldurs-gate-romance/
- Baldur’s Gate 2 Help: FAQ for SoA Romances and Other Headaches. (n.d.). Retrieved from http://www.bigorrin.org/amn5.htm
- Baldur’s Gate: Enhanced Edition-General Discussions- No, it’s not worth it. (2013, January 18). http://steamcommunity.com/app/228280/discussions/0/864957182873204516/
- Baldur’s Gate/Characters/Other Party Members. (n.d.). Retrieved from https://allthetropes.orain.org/wiki/Baldur’s_Gate/Characters/Other_Party_Members
- DanaDunchy. (2013, December 2). Baldur’s Gate 2: Hexxat Romance. YouTube. https://www.youtube.com/watch?v=r77i_5McUao
- DanaDuchy. (2013, December 1). Baldur’s Gate 2: Dorn (Romance & Quest). YouTube. https://www.youtube.com/watch?v=MKhHvcftsMU
- Dorn II-Khan. (n.d.). Retrieved from http://baldursgate.wikia.com/wiki/Dorn_Il-Khan
- Faren. (n.d.). Retrieved from http://www.spellholdstudios.net/ie/faren
- Hexxat. (n.d.). Retrieved from http://baldursgate.wikia.com/wiki/Hexxat
- Hexxat Romance Option. (2013, November). Retrieved from https://forums.beamdog.com/discussion/23972/hexxat-romance-option
- Johnson, L. (2013, November 27). A Semi-Gilded Lily. IGN.com. http://www.ign.com/articles/2013/11/27/baldurs-gate-2-enhanced-edition-review
- List of Baldur’s Gate Characters. (n.d.). Retrieved from https://en.wikipedia.org/wiki/List_of_Baldur%27s_Gate_characters
- MajorTomSawyer. (2006, November 11). BG Completed NPC & Romance List v3.1. Spellhold Studios. http://www.shsforums.net/topic/22836-bg-completed-npc-romance-list-v31/
- Mod (video gaming). Retrieved from https://en.wikipedia.org/wiki/Mod_(video_gaming)
- Nathaniel. (n.d.). Retrieved from http://www.spellholdstudios.net/ie/nathaniel
- Solaufein. (n.d.). Retrieved from http://www.weidu.org/main.html#install
- Wait…Are there same-sex romances now? (n.d.). Retrieved from https://forums.beamdog.com/discussion/23972/hexxat-romance-option