My S-corp received a distribution from an LLC of which it owns 50%. I cannot figure out how to categorize this deposit in quickbooks.

LLCs do not make distributions. They pay out Member Draw to a member even if that member is another LLC or an S corp.

The S corp would treat it as one of the following

stock purchase by the LLC

loan by LLC to S corp

but probably> temporary retained earnings to be applied against the pass through profit or loss from the LLC.  The S-corp gets a K-1 from the LLC for % of profit or loss to be applied against the P&L of teh S-corp