You would have to renounce that same-sex relationship if you were to become a muslim. It's simply not consistent with what the Qur'an says about living a life of purity. All throughout the Qur'an a story is told about the people of Lut [peace be upon him]. They practiced homosexuality and God destroyed them for it.
What happens after a genuine conversion [to Islam] is a fundamental change. You'll see Allah/God as being more beautiful than sin. You'll want Him more than you want homosexuality. We're spiritual beings at heart; we were made to worship Allah in all areas of life. That includes, but is not limited, to sexual purity. You and I were made to aspire to something greater than carnal lusts. A homosexual is not hopeless-- it's just the opposite. A person can find refuge at a moment's notice; that's one of the reasons the last two chapters of the Qur'an are prayers (to teach us to ask Allah to save us from the snares of this life).
I hope you will think about it and not write me off as a homophobe who hates homosexuals. In the rememberance of Allah do hearts find peace, the Qur'an says. What Allah offers is more precious. Selfless devotion to Allah is more sweet and precious than all the homosexuality in the world.