Retitling an account does not involve selling the assets owned. It simple means that you are changing the title of the ownership. It's basically the same idea as a newly married woman changing her account from her maiden name to here new married name or a newly married couple commingling assets in a new joint account. All cost basis information remains the same.
Some brokers will open a new account under the name of the trust and transfer the assets in from an existing account. Others simply change the title on the existing account, allowing you to keep the existing account number.
A simple call to call brokerage and asking them to change the title of the brokerage account to trust will not achieve this. You will need to provide them with a copy of the trust and then you will have to sign a form. In my experience, the majority of institutions only require the title page, the signature page and the page(s) that state the powers of the trustee. I have always resisted providing the entire trust because my will is none of their business.