The Shillelagh cantrip says that your attacks become magical, but the green-flame blade cantrip (SCAG, p. 143) says it works with a melee attack.
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.
Does the fact that my staff's attacks become magical stop green-flame blade from working?
The green-flame blade spell (SCAG, p. 143) states:
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 [...]
Here we see that it only requires "a weapon" not a "non-magical weapon". Similarly, its material component requirement is simply "a weapon". Thus we can conclude that it works with a magical weapon, but let's look at shillelagh just in case:
The wood of a club or quarterstaff you are holding is imbued with nature's power. For the duration, you can use your spellcasting ability instead of Strength for the attack and damage rolls of melee attacks using that weapon, and the weapon's damage die becomes a d8. The weapon also becomes magical, if it isn't already. The spell ends if you cast it again or if you let go of the weapon.
All this does is allow you to change the ability modifier added to melee attack and damage rolls, change the weapon die, and make the weapon magical. None of these would make the booming blade spell invalid as the weapon is still a weapon.
Note that both of these spells only affect "melee attacks" made with the weapon (in fact, making a ranged attack with a shillelagh-ed weapon will end shillelagh completely).
