I understand that I can't cast two spells per round (PHB p.202) even when using the Quickened Spell metamagic feat. However, is it allowable to cast a concentration spell like Crown of Madness on one turn, then on the following turn to use my main action to maintain that spell, and use Quickened Spell to cast another (non-concentration) spell with my bonus action?
Yes, you can do it
Maintaining the spell is not casting it.
On your subsequent turns, you must use your action to maintain control over the target, or the spell ends. Also, the target can make a Wisdom saving throw at the end of each of its turns. On a success, the spell ends.
Note that it says "use your action to maintain control over the target" - that's not casting a spell. If it was worded like "you must use your action to cast this spell again, without using spell slots", for example, you wouldn't be able to cast another spell as a bonus action in the same turn.
One simple way to see that you are not casting the spell again is that you don't expend any additional spell slots, even though the spell you would be casting is not a cantrip and it doesn't state you don't spend a spell slot (i.e. there is no specific rule in the spell telling you to go against the general rule - which would be to spend a spell slot).
As a note, the fact that it is a concentration spell doesn't matter. The only thing this changes is that the spell you are casting in the next round/turn must be non-concentration, as mentioned by you.
I understand that I can't cast two spells per round (PHB p.202) [...]
This is wrong.
You are not allowed to cast a spell as bonus action and then another (non-cantrip) spell as an action. (For clarification, you can't do it the other way around either, i.e., you can't cast a (non-cantrip) spell as an action and then cast another as a bonus action. For curiosity, you can't cast a (non-cantrip) spell as an action and a cantrip as a bonus action either. The cantrip has to be the action.)
You can, however, use more than one spell in your turn. For example, you can cast one spell as an action and another as a reaction, or use Action Surge (either multicass into fighter or EK) and use two spells as actions in the same turn. (Trivially, you can combine both the things I mentioned and cast three spells in a turn)
We need to break this down a bit.
I understand that I can't cast two spells per round (PHB p.202) even when using the Quickened Spell metamagic feature.
You can't consume two spell slots in a single round, but there's nothing stopping you from casting a spell as an action or bonus action, and then casting a cantrip as a, respectively, bonus action or action. So long as one is a cantrip, the other can be whatever.
cast a concentration spell like Crown of Madness on one turn, then on the following turn to use my main action to maintain that spell, and use Quickened Spell to cast another (non-concentration) spell with my bonus action?
Concentration spells don't require an action to maintain unless stated otherwise. They only require that you not cast another spell that requires concentration, and that you not encounter an effect which breaks concentration (usually damage). So if you cast another spell the next turn, regardless of whether it takes an action or a bonus action, the only stipulation is that it not require concentration. If it doesn't, then you're free to cast the spell and maintain concentration on your previous spell.
Spells which specifically require you to spend an action to maintain concentration have that precise condition: spend an action maintaining the spell, and the spell's duration is maintained. Performing a bonus action on top of that, so long as the bonus-action spell doesn't require concentration, won't break the original spell's concentration.