Can ForeFlight and Avidyne avionics talk to each other?

Yes. ForeFlight can connect via Wi-Fi to Avidyne's IFD 550/540/440 avionics to receive GPS position and transfer flight plans to and from an Avidyne with firmware 10.2 or higher.  ForeFlight can also receive AHRS (Attitude Heading Reference System) information from the IFD 550 to enable the full Synthetic Vision display. Additionally, with Avidyne firmware or higher, ForeFlight can receive ADS-B Weather and Traffic from Avidyne-supported ADS-B receivers via that same Wi-Fi connection.

This article has additional details about Avidyne connectivity.

You can read more about the ForeFlight Connect connectivity options with avionics providers including Avidyne here.