I answered this question a few months ago. In brief, the Spanish Inquisition was constituted in 1478 by King Ferdinand II of Aragon after lobbying the Pope intensely (Rome, in fact, seems to have been less than keen on the idea). Similarly as to how the Papacy used the Inquisition to exert control over its bishops, in Spain the "Catholic Monarchs" Isabel and Ferdinand used the inquisition to strengthen their authority over a peninsula rife with heresy. Perhaps as a reaction to increasing state control on religion, the Catholic Church established a centralized Inquisitorial Office in 1542.

How "unexpected" were investigations conducted by the Inquisition? We have a very famous example: 1611, Roberto Bellarmino, a Cardinal and important scholar in Rome, began preliminary investigations with regards to Galileo Galilei's theories on a heliocentric solar system, which if proven true begin to discredit the worldview as envisioned in biblical scripture. An inquiry as to his "heretical" past was also conducted by the local inquisitor in Padua, where Galileo had been studying the solar system. We don't know if Galileo was made aware of these initial investigations. However, the controversial nature of Galileo's work was well-known; by 1614 an ambitious Dominical preacher named Tommaso Caccini denounced several prominent scientists during a sermon in Florence's Santa Maria Novella church, and Galileo was one of those names. So it cannot have been a surprise for Galileo when the next year, Caccini deposited his accusations of Galileo's heresy (specifically, that he held that the earth revolved around the sun, tied to an accusation that his theory was born out of Galileo's lack of faith). In addition, the state apparatus of the Duchy of Tuscany, under whose patronage Galileo had been given a research grant at the University of Pisa without having any classes scheduled (a dream scenario for many academics) was well aware of the Inquisition having opened up a dossier on Galileo; the Tuscan ambassador to Rome knew all about the proceedings, and it is safe to say Galileo, who was frequently demonstrating experiments at the Tuscan Court in the Pitti palace, was similarly well-informed.

So we can safely say Galileo was expecting the Inquisition. In fact, you could argue that Galileo was provoking the inquisition, personally arguing his case in Rome and staying three months to appeal the decision. He was, effectively, attempting to change the minds of the inquisitors, exasperating the Tuscan ambassador ("His humor is set on reasoning with the friars and fighting with them with which he can't but lose ... Staying away from this country would be of great benefit and service to him" wrote back the Ambassador). But Galileo didn't stop; and would provoke the church as well as scholars at the Roman College until he was summoned by the inquisition a second time, in 1633. Again, it can't have been too much of a surprise, he had spent years picking fights with Catholic Scholars: instead of limiting himself to writing down commentary on the appearance of three comets in 1618, he wrote a rebuttal to the published observations of a scholar at the Roman College. In 1624, after an audience with newly elected Pope Urban VIII, Galileo petitioned him to accept the heliocentric model of the solar system. After the Pope said no, Gelileo published a lengthy treatise on the solar system anyway. Although only published in 1630 due to complications tied to his health the Church insisted on his presence, and in 1633 he was summoned before the Inquisition again. Now, his book had already been banned in 1632, reinforcing the assumption that Gelileo Galilei could have been reasonably expected to be expecting the Inquisition.