This can be stated in active form. Alternate translation: "Therefore the community of believers sent them from Antioch to Jerusalem" (See: [[rc://en/ta/man/translate/figs-activepassive]])
The abstract noun "conversion" means the Gentiles were rejecting their false gods and believing in God. Alternate translation: "announced to the community of believers in those places that Gentiles were believing in God" (See: [[rc://en/ta/man/translate/figs-abstractnouns]])
Their message causing the brothers to be joyful is spoken of as if "joy" were an object that they brought to the brothers. Alternate translation: "What they said caused their fellow believers to rejoice" (See: [[rc://en/ta/man/translate/figs-metaphor]])