The Inquisition operated from a1570-1820 in a building which later became the Senate and now houses the Museum of the Inquisition (Junin 548 -3 blocks from Cathedral).

To rid the area of Cathar belief
See en.wikipedia.org

An Inquisition tribunal was established in 1234. They settled in the city at the Tower of Justice. (wiki)
See fr.wikipedia.org

Monasterio de Santo Tomas. The Spanish Inquisition (Avila Branch) used to meet in this 15th century Gothic structure.

Pedro González de Mendoza, archbishop of Seville (and thus strongly associated to the cathedral) played a foundational role in the creation of the spanish inquisition. He cowrote the report (along with Torquemada) which moved the catholic monarchs to create the institution. In fact the Spanish inquisition was hatched in Seville while the catholic monarchs stayed in the city, likely in the alcazar.

El Castillo served as the seat of the Holy Inquisition until 1583

The Alcazar de los Reyes Cristianos was used as a forum and prison during the Inquisition. Part of it is called the Inquisition Tower

Palace of the Inquisition Plaza de Bolivar

Segovia was one of the headquarters of the Spanish inquisition. From 1486 to 1494 they stayed in the current house of the marquis of lozoya. Furthermore Torquemada, the first great inquisitor of the Spanish inquisition, was a prior in the convent of the holy cross in the city (also in the property)

"the Holy Office had its own ?palace?, which is now the Museum of Medicine on Republica de Brasil street"
See en.wikipedia.org

The Tribunal of the Inquisition was located in San Domenico and judicial procedures, sentences and often executions were held here.
See it.wikipedia.org