Fireball spell has no effect on Spike Growth.
The text of Fireball says:
It ignites flammable objects in the area that aren't being worn or carried.
However, the text of Spike Growth does not specify that the spikes and thorns created by the spell are burnt, damaged or if the suffer from any condition given by fire, independently on the ground from which they sprout from.
A similar case consists in the Insect Plague spell: a fireball exploding in the middle of the area of effect does not kill any of the bugs (see Are the insects from the Insect Plague spell affected by other damaging spells?).
As a counterexample, have a look at the Web spell:
The webs are flammable. Any 5-foot cube of webs exposed to fire burns away in 1 round, dealing 2d4 fire damage to any creature that starts its turn in the fire.
How can you negate the effects of Spike Growth?
Recall that the spell has a duration of 10 minutes and must be maintained via concentration.
There are several ways to try to negate the effects of Spike Growth. The following list is not exhaustive:
- Using Dispel Magic.
- Damage the caster to force them in making a Concentration saving throw.
- Inflicting to the caster the incapacitated condition (for example, via Sleep or Hold Person spells).
- Kill the caster.
- Fly over the area of effect (this negates the effects only for the character flying, not for everyone).
- Cast an Antimagic Field: in this way the caster of Antimagic Field can safely walk in the area of effect, and works for them and for everyone within 10 ft of them.
- Use a sledgehammer to crack a nut: use Wish (just for sake of completeness)
The description of each spell usually specifies what kind of interaction its effect has with other spells, if any. Of course, a DM could adopt particular and reasonable ruling: for example, a Fireball cast on the surface of a lake could produce vapor.
For the interaction between Fireball and Spike Growth, one may take inspiration from Neil Slater's comment to this answer:
A more reasonable interpretation of RAW (than taking damage from nothing) may be that whilst the spell is active dead bugs are replenished by the same magic that generated them in the first place. The spell is active and has a duration after all.
Hence, the spike and thorns are burnt by the fireball, but they are immediately replaced by new ones, created by the magic fueling the spell.