It is possible by taking a feat
As Dungarth explains, the range of the spell limits the caster to use the weapon attack on a creature within range of the spell.
The range of Green Flame Blade is 5 ft, and requires the caster to make an attack roll with your weapon, so even though the range of your weapon is 10 ft, the spell's restriction still applies. BUT, read the first bullet of the Spell Sniper feat:
When you cast a spell that requires you to make an attack roll, the spell's range is doubled.
That means that, if you take the Spell Sniper feat, a target is in fact in range at up to 10 ft, meaning that you can utilize any reach weapon with the spell. However, the secondary target of the spell must still be within 5 ft of the original target, as the restriction is not based on the range of the spell.
Update: Jeremy Crawford has confirmed on twitter that it works: https://twitter.com/JeremyECrawford/status/663511408269627392
Considering the below pages from the handbook sources I have, I have to say this sounds possible; though I didn't believe it at first.
Making an Attack: PHB pg 194
Choose a target: Pick a target from within your attack's range: A creature, an object, or location...
Resolve the Attack: You make an Attack Roll
First and foremost, you MUST make an attack roll. There are no sources to cite from either the PHB or the DMG that make exceptions to this rule. The recipient of any attack; melee weapon, ranged weapon, or spell attacks; does not have the option of getting hit without making the attacker roll first to get past armor class.
Now that we've dealt with that ... to make a melee attack you must choose 'A Creature within range' and since you are within your own range and count as a creature, you are a legal target for your own melee attack, unless the ability or spell says that you must target a HOSTILE creature. Making a general weapon attack does not carry that stipulation with it.
Green-Flame Blade is actually a spell with a range of 5ft that includes a melee weapon attack as part of the spell, and that it must be "A melee attack with a weapon against one creature within the spell's range.."
PHB page 204:
Targeting Yourself: If a spell targets a creature of your choice, you can choose yourself, unless the creature must be hostile or specifically a creature other than you.
So it seems that this cantrip can be targeted on yourself, AND the melee weapon attack that you must make as part of casting it can also target yourself, all under cited RAW.
The only two things you must overcome to successfully do this is first beating your own AC (Armor Class) with an attack roll as per PHB pg 194: 'Resolve the Attack', and second; surviving whatever damage your melee weapon attack roll does to yourself (and the damage of this melee attack get's stronger as your cantrip scales in power with your character level). So it's not an effective way to get an automatic, no-roll hit on an enemy.
Best Answer
There is no limit to the number of attacks you can make in a turn. There is a limit to how many actions and bonus actions you can make during a turn. Each action or bonus action that lets you make attacks will specify how many attacks it lets you make.
So, let's break down what you want to do.
Quickening the Green Flame Blade cantrip lets you cast it as a bonus action using the Cast a Spell (bonus) action. As part of this, you make one melee weapon attack. No part of this restricts what weapon you may use, other than a range of 5 ft.
You then use your action to take the Cast a Spell action, again casting GFB. As part of casting this, you make one melee weapon attack. Again, no other restrictions on weapon type.
The light weapon restriction comes from the two weapon fighting rule, which dictates making an attack as a bonus action after using the Attack action. Since you didn't use the Attack action, this rule has nothing to do with what you did.
Yes, you can do this while wielding a single weapon.