I’m not super knowledgeable but since no one else has responded I figure I can provide better input than nothing. Generally, bluetooth devices use 2 different audio profiles for calls and for normal listening. Most likely your device used the A2DP profile when you are listening to music or otherwise just using your device for audio only purposes. This profile has decent quality. When you connect to a call however, this profile does not support send/recieve audio, so bluetooth has to use a different profile. Usually that profile is lower quality, and you may have to manually change this profile. Usually there may be a “Headphone/Headset” option that you can choose. Headphone is A2DP, and Headset is HFP (hands free). You can read more about them on the “Bluetooth profile” wiki or just google it. There are hacky solutions online that let you change these profiles with a single button if the call program does not automatically switch you over.
I can’t send links unfortunately, but search “How to set up automatic connection of bluetooth headset” online and you should find some potential solutions for getting the device to connect automatically. I wouldn’t bet on it working though, bluetooth in general is just a really frustrating technology.