Can I use Sabrina's Suggestion to copy an opponent's Green's Exploration in their hand if I have a Pokemon in play with Abilities?
Nope sorry, you still have to obey the restrictions on the copied Trainer card. Since you have a Pokemon with Abilities in play you cannot use Green's Exploration even from your opponent's hand.
Source: TPCi Rules Team (2019-09-12)

Relevant cards

Note: All applicable cards may not be listed below.
SM10_175
SM10_175
SM10_209
SM10_209
SM9_154
SM9_154
SM9_181
SM9_181