Level Select: (source: http://www.gamewinners.com/SMS/Quartet.htm)

At the title screen, press PAUSE 12 times, then hold UP/LEFT and press buttons 1 and 2 at the same time on controller 2.  This only works in two-player mode.

Sound Test: (source: http://www.gamewinners.com/SMS/Quartet.htm)

At the title screen, press PAUSE 4 times, then press button 1 on controller 2.

Larger Beam: (source: http://www.gamewinners.com/SMS/Quartet.htm)

At the title screen, press PAUSE 14 times, then press button 1.