The SparkFun MIDI Shield board gives your Arduino-based device access to the antiquated, but still widely used and well supported MIDI communication protocol, so you can control synthesizers, sequencers, and other musical devices. The MIDI protocol shares many similarities with standard asynchronous serial interfaces, so you can use the UART pins of your Arduino to send and receive MIDI’s event messages.
With some adapting, most likely, but you would not be able to use this as a shield on top of the Q.
The UNO Q uses 3.3 V I/O with some 1.8 logic pins. Whereas this unit was designed around 5 V UNO boards. Arduino specify that there Q’s are not 5V logic tolerant so mixing the two without some modifications to handle the voltage change would not be wise.
That said, with some tweaking and voltage shifting, you should be able to get some capability between the two units.