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. lf 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 conjuction with Green-Flame Blade.
Compare this to the spell Shadow Blade (from Xanathar's) 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 weapons properties.