"We wish to enjoy the same social privileges and contractual rights ... and not to be treated as we are being treated as second-class citizens," she said.
Conde is suing Puerto Rico's heath secretary, who oversees the island's demographic registry, as well as the registrar of vital records.
Conde said the lack of recognition of their relationship complicated things when her young daughter had to have open-heart surgery for a second time and Alvarez could not participate in the decision-making process.
"Gay and lesbian individuals have suffered a long and painful history of societal and government-sponsored discrimination," she said in the lawsuit.
Currently, 17 U.S. states and the District of Columbia allow same-sex marriage.
It is the first time that someone has filed a lawsuit of this kind in the jurisdiction of Puerto Rico, where lawmakers last year approved four measures in favor of the gay community. One of the bills signed by Gov. Alejandro Garcia Padilla prohibits employment discrimination based on gender or sexual orientation, and another extends a domestic violence law to gay couples.
The extension of the domestic violence law places Puerto Rico ahead of 46 U.S. states that have no such law, according to local human rights activist Pedro Julio Serrano.
Puerto Rico also is ahead of 33 U.S. states in approving a hate crime law, he said.
"People always think that Puerto Rico is behind," he said in a phone interview. "But we're actually ahead of the majority of U.S. states."
However, certain gay rights are still being debated in Puerto Rico.
Last year, the island's Supreme Court voted 5-4 to uphold a local law banning adoptions by same-sex parents. The decision was unsuccessfully appealed by a Puerto Rican woman seeking to adopt a 12-year-old girl that her partner of more than 20 years gave birth to through in vitro fertilization.
The judges said it was up to legislators to change the adoption law, and a lawmaker earlier this month filed a measure seeking such a change.