To keep cost basis information and other information about the stocks, you may want to try "transfer shares" between the two accounts after you have set up the new account. The old transactions will stay in the old account. You must transfer the stock "as securities" between the brokerages (not as cash with a sell and buy transaction), and in Quicken you use "transfer shares". New transactions will be recorded in the new account. Verify that the cost basis information for each stock/security is computed correctly in the new account.
Are the stocks ESPPs?
IF, you're moving EVERYTHING our of the present/old account ... just deactivate that account from downloading, rename it to the new name, and reactivate it.MUCH easier than "moving" anything.