"For a trust, do I input any capital gains that I allocate to income to a beneficiary as a first tier distribution?"
No, because according to the facts you previously set forth you, as trustee, have the discretion as to whether or not to make distributions of capital gains to a beneficiary.
Firs tier distributions only include income that is required to be distributed currently, whether distributed or not.
See https://www.irs.gov/instructions/i1041#idm139674712043984
