I am an attorney. Is there a way to track monies going in and out of my IOLTA trust account without it counting as income twice?

Good day, @karen71.

If a transaction is double-counted, you have to check the categorization of the entry. Make sure to tag them correctly as Expense for money out and Income for money in. 

I'm adding a screenshot of where you can edit the category in a transaction.

Use Transfers for both transactions when moving money between accounts. I recommend checking out this article for more information: Transfers between accounts in QuickBooks Self-Employed. Though the article is for the Community Canada website, much of the information and steps work for the US version. 

Mention me in the comment section if you need further assistance. I'm always here to help. Have a fantastic day!