Bigby's Hand counts as an object, but if it is adjacent to an enemy who then moves away, can you use your reaction to take an opportunity attack?
No - objects created by spells only have the abilities granted to them in their descriptions
The hand is an object created by a spell effect and is not granted the ability to take an opportunity attack
Unfortunately, Bigby's Hand is an object ("Large hand of shimmering, translucent force") but not a creature. Unlike creatures which have more general rules concerning their abilities, spell effects only have the abilities granted to them by the spell description.
When you cast the spell and as a bonus action on your subsequent turns, you can move the hand up to 60 feet and then cause one of the following effects with it.
All of the abilities of the hand thus require you to spend a bonus action on your turn to enable. One of these abilities gives the hand the ability to grapple (an ability that all creatures have by default). It would need a similar ability spelled out that gives it the ability to take opportunity attacks.
The spell says nothing about granting the hand the ability to take (or provoke) opportunity attacks so it does not get that ability.
The hand does not meet any of the requirements needed to make an OA anyways
Additionally, looking at the description for opportunity attacks yields several more reasons why this would not work.
You can make an opportunity attack when a hostile creature that you can see moves out of your reach. To make the opportunity attack, you use your reaction to make one melee attack against the provoking creature. The attack occurs right before the creature leaves your reach.
In order for the hand to be able to take opportunity attacks it must meet all the requirements above. However:
- The hand cannot see
- The hand does not have a reach
- The hand does not have actions or reactions of its own
- The hand does not have the capability of making melee attacks on its own