Sharon - This is a general and typical answer. I also need to state that I am not an accountant and I have not seen the trust. My answer then is maybe. The trust would only pay income tax on income the trust received. The beneficiaries of the trust may also have to pay income tax on the income they receive. There are a few issues that would need to be clarified to help you understand the exact situation. Type of trust, qualification of assets in the trust, type of assets in the trust, and finally the wording of the trust as to income and distributions. Keep in mind too that there are many, many types of trusts with infinite variations in their wording.
If your concern relates to the trust paying taxes on some income from a holding in the trust and then the beneficiaries having to pay income tax on the very same income then no is the answer. Income can only be taxed once in a trust to beneficiary situation. Think of the trust as an entity just as a person. The clarity relates to which entity received the taxable income and that entity incurs the tax liability.
Good information from Augustus. I'd just add a thought, maybe more from a "planning" standpoint....
Trusts that exist as a separate taxable entity pay tax on income they retain but not on income that's distributed to the beneficiaries. The beneficiaries pay income tax on that.
Federal income tax rates are very punitive for Trusts. For example, a Trust will pay the highest 39.6% federal income tax rate on taxable income over $12,150. (Yes, you read that correctly.) On the other hand, a beneficiary will not pay a 39.6% federal income tax rate until their taxable income exceeds $406,750.
Depending on the stipulations in the Trust document, beneficiary needs, etc., it may make sense for the Trustee to try to distribute ALL of the Trust's taxable income each year so that net income taxes will be lower in the hands of the beneficiaries.
Just a thought. Hope that helps.