Let's look at what you're asking here:
In summary, I want to implement a house rule where a player can use green flame blade with a single available opponent, applying the complete damage to that opponent. Would this be gamebreaking if the player has to use a wish spell to enable the change?
It seems like you have 2 possible approaches: simply implementing this change as a house rule, or allowing a player to use the wish spell to change the rules.
What if you implement this change as a house rule?
If you look here you can find where someone has calculated the average damage for a wide variety of melee attacks & cantrips. Against a single target the green flame blade cantrip is slightly better than just hitting with a martial weapon. Against two or more targets the green flame blade is almost as good as a pure fighter with all their extra attacks. Your player is essentially asking to do almost as much damage with their attacks a pure fighter in all (melee-range) situations, plus keep all of their other spells and class abilities that a fighter doesn't get.
That's a very strong option, and if you have another player with a fighter character then you should expect them to be jealous. But even if you're OK with that aspect of the change, another aspect to consider is the distorting effect it has on other classes. Any class that wants to hit targets in melee but does not get extra attacks (most notable rogues, but also some other classes & archetypes), almost has to pick up this cantrip; it just offers too much single-target damage to not be using it instead of a regular attack.
In short, this won't be game-breaking, but it will definitely be game-distorting. Proceed at your own risk.
What if a player implements this change via the wish spell?
First, most of the above concerns apply. If the change is only for the caster of wish then that's a straight-up permanent buff to that character that nobody else receives. If the change is to everyone then it has the additional distorting changes mentioned above. The real issue, though, is the precedent set by allowing a player to use the wish spell to change the mechanics of the game.
While the books for D&D don't really delve into this very much, one broad approach for understanding roleplaying games in general is to separate play into two "layers": the "fiction" layer is the world that the GM & players are collectively describing, while the "game" layer is the set of rules the game system provides for determining the outcome of uncertain events in the "fiction" layer. When Anna describes her orc barbarian, Grogak, striking a goblin with his greataxe, that description is in the fiction. It is then translated down into the game rules as an attack. Anna rolls the attack, hits, rolls damage, and Bob the GM tells her that the goblin has been reduced to zero hit points. That result is then translated back up into the fiction layer to a description of Grogak striking the hapless goblin and cleaving it in two in a shower of blood & entrails.
Allowing a player to use the wish spell to change the fiction layer is fine; that's what the wish spell is for, making changes to the fictional world. Allowing a player to use the wish spell to change the rules layer, however, is deeply dangerous. Allowing a player to do so, regardless of how tame the wish, sets a precedent that this sort of thing is allowed. While your player has chosen a relatively mild rules change (though not one without consequences, as described above), there are other much more dramatic rules changes that could be wished for (and some players have wished for, as can be seen on this site), and if you allowed the small change then players will argue that you should also allow larger changes, and then still-larger changes, and so on. If you give a mouse a cookie...
The text of the wish spell explicitly encourages GMs to twist wishes that don't draw from the set list of reliable effects (and I think we can all agree that what your player is asking for is much stronger than anything on that list). In general I would recommend simply not allowing wishes to change the rules layer, but if you do choose to allow it then you should strongly consider twisting the change so that it has a downside that discourages players from making such changes. In cases where players are asking for even more powerful changes, you might even twist it such that result of their wish is entirely bad (though I feel that would be overkill in your current case).
In this case you might modify the spell so that when the character deals the second damage to the same target as the first damage, the character also takes the second damage. The player thus has that ability to focus the spell's damage on a single target as an option, but it's not an option they'll be choosing when they don't really need it. You might even describe it as reality twisting to punish them for making such an ambitious wish (after all, in the fiction layer they've effectively wished for a change in how magic itself works).