If a spell caster is inside a globe of invulnerability and they try casting a spell, can somebody outside of the globe use counterspell to block their attempt to cast that spell? I had though it would not be possible, but the globe of invulnerability only protects creatures and object while counterspell is arguably targeting the spell itself.
No, it can't
Counterspell "interrupt[s] a creature" - if that creature is within a Globe of Invulnerability then Counterspell "has no effect on them."
Nothing inside the globe can be affected by Counterspell even if you cast it at higher levels
Although your exact question has not been answered on Sage Advice, I believe the answer concerning how Dispel Magic applies to the Globe, applies just as well:
dispel magic could affect the barrier created by globe of invulnerability, but not anything inside the barrier
which aligns with the description of the Globe:
Any spell of 5th level or lower cast from outside the barrier can’t affect creatures or objects within it, even if the spell is cast using a higher level spell slot
So you're better off trying to use Dispel Magic to get rid of the Globe first.