I understand that the colonisation of South America was at a similar to to the Spanish Inquisition where Muslims and Jews were forced out of Iberia. Is there any evidence to suggest many of them travelled to South America?
The first thing to address here is that the Portuguese and the Spanish Inquisitions had very different trajectories and prerrogatives from one another. The Spanish crown, interested both politically and spiritually on collaborating with the papacy, cooperated extensively with the Inquisition since the last decade of the fifiteenth century. As a result, there eventually were actual, stablished inquisitorial tribunals in Mexico and Peru, the two of them with full prerrogatives.
The Portuguese crown, on the other hand, had a more ambiguous relationship with the Inquisition's tribunals, as they were seen as papal tools to undermine the king's authority and viabilize the Pope's interference in the country. The end result was that the crown and the Inquisition disputed for influence over one another, and even if the former was eventually stablished in Lusitan lands in 1536, its relationship with the political authorities was always rockier than in Spain.
To answer your question, though, all this previous information means that migratory fluxes of Jews to Spanish colonies in America were very different from Portuguese ones. As your question mentions South America specifically, though, there is evidence not only to suggest, but to affirm without doutb that there were many Jews and new christian conversos who went to the continent, specially to Portuguese America's Captaincies of Pernambuco and Baía de Todos os Santos, mostly between the sixteenth and seventeenth centuries.
The first factor that can explain the migrations is the legal discrimination Jews and new christians conversos suffered in Europe in general. In Portugal specifically, there were statutes of law issued specifically for them, denying them certain rights given to those "old Christians of pure race" (which at the time meant coming from a bloodline of christians that didn't have muslim, pagan or jewish ancestors, nothing to do with the concept of race elaborated by eugenists in the nineteenth century). This means the new Christians were prime targets for the Inquisitorial efforts, and this added to the climate of official persecution they had to endure. This culminated with successive royal banishment decrees (from 1497 up to the end of the seventeenth century) for the Jews who refused to convert to christianity. Not only that, but social discrimination was also common, and antisemitic aggression wasn't unheard of, sometimes erupting in pogrom-like behavior, such as happened in Lisbon in 1506.
Those various forms of social and institutional violence made for a good incentive to move away. Those who could went either to other European nations such as the Netherlands and France or to the Portuguese colonies in America. There, they had a chance to start anew, in a way, without completely severing their network of contacts within the Portuguese Empire or losing touch with their families.
Stuart Schwartz also argues there was a social climate of more extensive tolerance in America than in Europe, due to many factors that range from the need to have a stable society in order to survive in a harsh new land to there not being any official inquisitorial tribunal in the Portuguese American colonies. Even if this is a point of contention among specialists, even the detractors can't argue there was less persecution in America, which made it a potentially desirable destination for the Jewish population.
The jewish presence in Portuguese America is heavily documented both during the visitations by Inquisitorial officials between 1591-1595, 1618-1621 and 1627-1628; and during the Dutch invasion of Pernambuco (1630-1654), when jews were allowed to practice their religion freely, had the limitations of their rights mostly removed and even built the first synagogue in America. This is not to mention political and administrative communication from all over the Portuguese colonies, which have helped show how wealthy new christians were very capable of rising to the ranks of the colonial economical elites.
That said, jews and new christians still experienced social and legal discrimination in Portuguese America, which was responsible for either their successful integration in christian society and the subsequente abandonment of jewish heritage as a means of legitimizing themselves in the eyes of society - a strategy favored by those economical elites, who sought to marry christian women from old christian families; or their leaving the Portuguese colonies to look for better places to be, a movement that was very noticeable after the Dutch left in 1654 and the old laws and customs were brought back in full force.
As for muslims, though, it's much more common to find them among the yoruba slaves known as malês - the yoruba word for muslims. Those were africans from multiple ethinic groups trafficked to the Captaincies of Pernambuco, Rio de Janeiro and Baía de Todos os Santos, and were much more numerous in Portuguese America than moors or people of moorish descent. Not that it was impossible to find them, but mostly due to the long-standing animosity between the kingdoms of Spain and Portugal and the moorish caliphates that remained after the Reconquista, there were no significative migrations of moorish people to the Iberian colonies in America, and the ones who came also tried their best to integrate and not stand out so as not to be persecuted. As for the malês, they eventually rebelled against their masters, but were defeated and from then on, much more heavily controlled, both physically and culturally.
Bibliography
"All Can be Saved", by Stuart B. Schwartz.
"Cristãos Novos na Bahia", by Anita Novinsky.
"História da Inquisição Portuguesa, 1536-1821", by Giuseppe Marcocci and José Pedro Paiva.
"Jerusalem Colonial - Judeus Portugueses no Brasil Holandes", by Ronaldo Vainfas.
"João Nunes, Um Rabi Escatológico na Nova Lusitânia", by Angelo Adriano Faria de Assis.
"Os Judeus no Brasil. Inquisição, Imigração e Identidade", by Keila Grinberg (org.).
Edited to correct a name on the bibliography.