Quartet has a bonus round?

I haven't really had the chance to play this that thoroughly. Have you check gamefaqs? They may have a walkthrough on there, though I doubt it.

Is the bonus round the one where you shoot enemies to get up to 5000 points at a time?
If so i've found 2 bonus levels,if you've read the gameFAQS.com description on how to get to it it's not very helpful,i found their walkthroughs aren't detailed enough so it didn't help me either when i was stuck on Golden axe warrior last week.
I'll end a 17 year mystery for you,
the first bonus screen is on stage 2 basically walk along the bottom of the screen to the far end then jump up to the door then in the interior level walk along the bottom again and kill that lizard and make sure you have the jet pack when finished,go collect the star and bonus point ball.Exit the level you came from and keep pushing button 1 until you're in the bonus room,you will need the jet pack though and if you get hit you lose it altogether so keep dodging.
That one i found myself, it's not on gameFAQS.
For the second one it's on level 5,however you must go and kill the boss first (not explained on FAQ) then go back onto the exterior level with the jet pack and go underneath the door you came from where that small platform is in the very bottom right corner,you must shoot just underneath it,and it looks like you're hitting something invisible,hit it 10 times or so until it disappears.It must be hit from the right hand side (also not explained on FAQ) If you try from the left i think it goes on forever.Then head back up to the door and keep pushing button 1 a few times until you enter the bonus room.

That level select works for a 1p game also,just push up on pad 2 and hit pause 12 times on the title screen then press button 1 on pad 1.

It works! YOU RULE!