Do you have a particular caliphate in mind? There have been quite a few.

Hunayn ibn Ishaq al-Ibadi (808/809-873), also known as "Joannitius", was a Nestorian translator, writer, and court physician to Caliph al-Mutawakkil. According to the Encyclopaedia of Islam (Brill), Hunayn was "the most important mediator of ancient Greek science to the Arabs" and renowned for his prolific translations of Greek medical texts, notably Galen and Hippocrates, to which later Arabic physicians would be indebted.

Hunayn's family was originally from al-Hira and had stayed faithful to the Syrian Nestorian church. He was bilingual - fluent in both Syriac and Arabic - and later picked up Greek after his expulsion from the House of Wisdom in Baghdad. As the story goes, after a dispute with his teacher, Hunayn disappeared to either Alexandria or Constantinople; two years later he returned fluent in Greek, made up with his teacher, and was encouraged to translate Greek texts. His method in translation was sophisticated, avoiding literal word-for-word translations and basing the translations on the collation of numerous different sources, often travelling to Syria and Egypt in search of texts. He translated Greek into both Syriac and Arabic; Syriac for his fellow Christians doctors and Arabic for his patrons. Aside from being a translator, Hunayn was also a prolific writer on diverse subjects, notably with a special interest in opthalmology.

How Hunayn came to become physician to Caliph al-Mutawakkil is recounted in his auto-biography. According to Hunayn (ahem), he was the subject of a plot by envious Christian colleagues (and the head of the Nestorian church) to denounce him as an atheist/heretic and to have him punished by the caliph. The plot was hatched by Bakhtīshū‘ ibn Jibrā’īl, another Syriac Nestorian physician, and involved tricking Hunayn into spitting on an icon of the Madonna in the presence of Caliph al-Mutawakkil. Having fallen into the trap, Hunayn had his estate confiscated and was imprisoned + tortured for half a year by the Caliph. Facing execution, he was only released after a "miracle" where al-Mutawakkil fell seriously ill and was visited by Jesus in his dreams telling him to let Hunayn treat him. After Hunayn cures the Caliph of his ailments ("the medicine had moved his bowels three times, and he felt the onset of recovery"), Hunayn is released, rewarded many riches, and made court physician, etc.

Also worthy of note is Hunayn's son Ishaq (ibn Hunayn), who was himself an important translator and is best known for the third Arabic translation of Euclid's Elements. In the Sir Thomas Heath editions of the Elements it is asserted confidently that Ishaq made the translations directly from Greek, however elsewhere the translation is sometimes attributed to Hunayn. It is possible that Hunayn first translated the text into Syriac and his son then translated the Syriac into Arabic.