This works, but you are making 3 separate attacks. You also could do more on round 1
Round 1 is rather standard, all you're doing is casting a spell, but that's all you're doing. You have an available bonus action and also an action granted by haste that you simply aren't using (or haven't mentioned using).
Round 2 is somewhat complicated but works out perfectly well; the only rule to be wary of when casting multiple spells is the following from the "Casting a Spell" section:
[... If you cast a bonus action spell,] you can't cast another spell during the same turn, except for a cantrip with a casting time of 1 action.
What you're doing here is casting booming blade (a cantrip) as a bonus action using the Sorcerer's Quickened Spell Metamagic, and then casting green-flame blade (a cantrip) as an action, using your regular action.
Quickened Spell will work perfectly well on cantrips, so this part works out:
When you cast a spell that has a casting time of 1 action, you can spend 2 sorcery points to change the casting time to 1 bonus action for this casting.
You still have haste's additional action available, and you have decided to use it on taking the Attack action, and using Divine Smite with that action. Divine Smite states:
Starting at 2nd level, when you hit a creature with a melee weapon attack, you can expend one spell slot to deal radiant damage to the target, in addition to the weapon’s damage. The extra damage is 2d8 for a 1st-level spell slot, plus 1d8 for each spell level higher than 1st, to a maximum of 5d8. The damage increases by 1d8 if the target is an undead or a fiend, to a maximum of 6d8.
Just because you are expending a spell slot here does not mean you are casting a spell, so no part of the bonus action spellcasting rule will invalidate this action. Note that you could also use Divine Smite on either (or both) of the blade cantrips as well.
The cantrips are not like the Paladin's smite spells; their attacks are made immediately when casting the spell
Let's just look at booming blade:
As part of the action used to cast this spell, you must make a melee attack with a weapon against one creature within the spell's range, otherwise the spell fails. On a hit, the target suffers the attack's normal effects, and it becomes sheathed in booming energy until the start of your next turn. If the target willingly moves before then, it immediately takes 1d8 thunder damage, and the spell ends [...]
This spell requires you to immediately make a melee attack when you cast the spell, it cannot be held onto or stored like other Paladin spells such as branding smite:
The next time you hit a creature with a weapon attack before this spell ends, the weapon gleams with astral radiance as you strike. The attack deals an extra 2d6 radiant damage to the target, which becomes visible if it is invisible, and the target sheds dim light in a 5-foot radius and can’t become invisible until the spell ends.
Note that their wordings are entirely different, booming blade's attack is made during the action used to cast the spell, whereas branding smite doesn't grant you an attack at all, instead it only activates when you make an attack.
Having separate attacks is usually better
You're making three separate attacks on your turn instead of one giant attack; thus you have more options for how you want to distribute the damage, and a single miss is far less punishing (though a critical hit is also less rewarding). You can also move in-between the attacks if you wanted, allowing you even further customization and targeting options. It even lets you apply Divine Smite more than once if you so choose.