How can I upgrade my clients own existing subscription(Essential) to Plus (accountant a/c) ? Will it effect my connected banks and entries already posted ?

It will not affect your existing data after upgrading your plan. Bear in mind, no promotional rates apply.

https://quickbooks.intuit.com/learn-support/en-us/help-article/wholesale-discount/transfer-clients-proadvisor-preferred-pricing/L4GT5C1J2_US_en_US

Another option, open a new blank QBO Essential with your Preferred Pricing and utilize the Copy tool to duplicate data from their old account.

https://rewindio.grsm.io/copy