No, it doesn't.
In fact, you can't even use a Flame Blade spell as part of an Attack (action).
The spell states:
You evoke a fiery blade in your free hand. The blade is similar in size and shape to a scimitar, and it lasts for the duration. If you let go of the blade, it disappears, but you can evoke the blade again as a bonus action.
You can use your action to make a melee spell attack with the fiery blade. On a hit, the target takes 3d6 fire damage.
The only mechanical uses of the the fiery blade are its unique action (make a melee spell attack, deal 3d6 fire damage on hit), and its ability to shed light.
Since you can't make a melee weapon attack with it, it isn't a weapon, and thus can't be used in conjunction with Green-Flame Blade.
Compare this to the spell Shadow Blade (from Xanathar's Guide to Everything, p. 164) which says:
You weave together threads of shadow to create a sword of solidified gloom in your hand. This magic sword lasts until the spell ends. It counts as a simple melee weapon with which you are proficient. It deals 2d8 psychic damage on a hit and has the finesse, light, and thrown properties (range 20/60). [...]
The spell description specifically states that the effect counts as a weapon, and it defines the weapon's properties.