"Today marks the next major phase in our pursuit of accountability for the assassination," Quiñones said in a statement posted on social media.
The 53-year-old president was shot dead in July 2021 at his private residence in the hills above Port-au-Prince, a killing that left a gaping political vacuum in the Caribbean country and emboldened powerful armed groups in the country.
The killing has prompted multiple investigations and indictments in Haiti and the United States.
On Sunday, Haitian Justice Minister Patrick Pelissier confirmed in a statement that suspects had been transferred from Haiti to the US, adding that Haitian judicial proceedings had been delayed.
"While the American procedure has recorded significant progress, notably arrests, prosecutions, and convictions, the Haitian procedure remains confronted with major difficulties and has not, to date, resulted in a trial being held, after five years of proceedings."
US prosecutors have argued that suspects in the case used South Florida as a "staging ground for a deadly foreign plot".
Thirty people have been indicted in the US in relation to the crime, according to Quiñones' statement on Sunday.
In May, four South Florida men were convicted of plotting to kill the Haitian president by hiring mercenaries to assassinate him at his Port-au-Prince home.
Prosecutors argued during the nine-week trial in a Miami federal court that the men assembled two dozen former Colombian soldiers and supplied them with money, guns, ammunition and tactical vests in a conspiracy to kill Moise.
In Haiti, prominent human rights activist Pierre Espérance said that the slow pace of the Haitian case and the transfer of the 18 suspects to the US was a "disgrace for the Haitian justice system".
He argued for strengthening the country's justice system to allow for such complex cases to be tried inside Haiti, rather than sending them to the US for prosecution.
