Welcome to Android Central! There may be a couple of possible solutions:
1. First, go to
Settings>Bluetooth on the phone and tap the gear icon next to the car system. Make sure Contact Sharing is turned on. If it already is, then try forgetting the phone from the car and the car from the phone (in the Bluetooth settings), then go through the pairing process again. After it pairs, watch the phone screen closely and see if it asks you for permission to share contacts to the car.
2. If solution
#1 doesn't work, then open your Contacts app and tap Menu. Look for the option to Customize which contacts are displayed. Instead of selecting Show All, select your contact groups individually -- this was how I had to do it on my previous car, since "Show All" would inexplicably not show any contact in the car.