Yes, diplomatic exchange between Rome and India occasionally pops up in Imperial rhetoric as demonstrations that the Indians were submitting to the imperial might of Rome, or paying homage to the glorious Roman emperor, but this is of course simply rhetoric. There was a brisk trade between Rome and southern India carried on through the Egyptian Red Sea ports, and Indian merchants seem to have been resident in Alexandria. It is not unlikely that there were occasionally diplomatic exchanges relating to this.

The best discussion of this is in Grant Parker's Creation of Roman India.