Lady Quark, aka Tashana, was originally from Earth-6, a place where the United States had lost the Revolutionary War. Her husband was Lord Volt, and their marriage was an arranged one. She preferred women, and he preferred men, but they agreed to the marriage for the sake of their world and produced a child, a daughter they named Liana. They ruled together as a monarchy and loved their child unconditionally,despite their distaste for one another.
And then came CRISIS ON INFINITE EARTHS. As anti-matter spread throughout their world, destroying it, Lord Volt and Lady Quark defended their home. However, Lord Volt was killed trying to save their daughter, who was also killed. The entire planet was destroyed, but Lady Quark herself was rescued by Pariah, a dimensional traveler. Once the Crisis was over, Lady Quark joined Pariah and Harbinger in exploring the new, sole Earth that had been created. She is eventually recruited by Vril Dox to join his L.E.G.I.O.N., a peacekeeping space force, and is seemingly killed in a later mission by a parasitic alien shapeshifter. However, she apparently survived the encounter and was captured by Alexander Luthor of Earth-3, who imprisoned her in his tower as a part of his plan to bring the core Earths back into existence. She soon escaped and joined Earth's heroes in defeating Luthor once and for all.
Because of the massive changes wrought by the Crisis, Lady Quark could no longer be from Earth-6. However, it was never revealed where she originated, but one can assume it was from some alternate dimension.
Thanks to FLASHPOINT, Lady Quark has been reintegrated into the current multiverse as a resident of Earth-48, a fusion of her original Earth-6 and pre-Flashpoint Warworld/Earth-48, home of the Forerunners. She, Lord Volt, and their daughter Liana are once again members of the royal family, alongside Viza'aziv, who had been the sole remaining Forerunner of pre-Flashpoint Warworld.
At some unrevealed point, Lady Quark ended up in the custody of A.R.G.U.S., but she was freed by the Justice League of America member, Vibe, during a mass breakout and has presumably returned to her homeworld.