Remember, you are trying to get the best sound you can from your Quatros, not to just use tubes.
Quatros have great advantages to getting tubes to sound good;
1.0 Super easy impedance.
2.0 Main amplifier isn't really driving the bass, a tube weak spot.
A sore spot is Quatro's already have a fixed "house" sound that is warm and nice but MORE of that sound can get almost too warm with tubes. It's the old more of a good thing is not good thing. It takes gutsy and expensive tube amps to sound crisp and clean. Have you considered a tube stage preamp instead? This is an easier option with Quatro's. Look at, and audition, an Audio Research LS-17. This would probably be nice into a SS amplifier. This unit should not be too "warm" into Quatro's.
So for a given set-up, Quatro's can do well with tubes, don't get me wrong on that. BUT, you want to get the sound as good as you can for the MONEY, not just use tubes, per say. Getting an inexpensive tube amplifier to sound good, but worse than a SS amp in the same price point, seems counter productive to the end game, getting the sound as good as you can. You might switch your effort to tube pre and SS power to maximize your end game.
But this is indeed entertainmnet, so LISTEN before you buy to get what you enjoy. Who cares what we "think"!