De Quesada was appointed Governor and intendant of Honduras in 1783, and remained in this office until 1789, when he was replaced in the Honduran government by Alejo García Conde. The following year, in 1790, he was appointed Governor of East Florida by Charles IV of Spain to replace Vicente Manuel de Céspedes and assumed the governorship in July 1790, during the Nootka Sound Crisis. After his arrival at Saint Augustine, as a consequence of the Nootka Crisis, De Quesada revamped the defenses of the city, which were not extensive enough, according to him and his engineer, Mariano de la Roque.
The garrison was small and dispirited, as the Spanish royal subsidy (Situado) had not returned to Florida since 1787.
Eventually, De Quesada obtained also the titles of Vice Royal Patron, and Subdelegate of Saint Augustine. In 1793, under his administration, began the Construction of the Cathedral Basilica of Saint Augustine in East Florida, not ending until 1797, a year after that Nepomuseno left the government of East Florida.
In addition, De Quesada banned the recognition of runaway slaves who had fled from the British colonies to Florida seeking freedom. He became ill in February 1796, and continued to serve as governor of East Florida only until March that of year, when he was succeeded by Bartolomé Morales.
He died in 1798.