1. If MCP4822 can work also with 3,3 V, you can use it with STM32 MCUs.
If you use default pins of BluePill-board, SPI should work.
2. If you want to program this DAC with ArduinoIDE, you will need STM32duino and for upload STM32CubeProgrammer.
BTW: your subject "no audio" sounds like you already uploaded a sketch to STM32F103.
Do you have a sketch or not?
If you have one, you should show it!
Mozzi is a library? Normally libraries are installed inside ArduinoIDE with library manager.
3. Which MCP4822-library you used for Arduino Nano?
No information about from you!
May be it will work also with a STM32F103.
And why you don't use a STM32, that has DAC inside???
Two years ago was a similar topic here and a link to a library:
https://github.com/SteveGdvs/MCP48xx/tr ... 9001970323
Otherwise ST has published something, you should type in "MCP4822 STM32" at Google (there is no real link). It was made for Keil and a NUCLEO-STM32L476RG board, but there is a definition for a sine wave table inside code.