I am not sure what you are asking.
Thunderbird is a free, open-source, cross-platform application for managing email, news feeds, chat, and news groups. It is a local (rather than a browser or web-based) email application that is powerful yet easy-to-use.
This application uses it as a type of 'channel' to add emails too from gmail, outlook, yahoo, aol etc.
Google is just a search engine.
Are you trying to put thunderbird back on your cellphone and filter in another email account like gmail, outlook etc?
Did you already have thunderbird on your phone and want to use it as your default?
Please answer the questions above and if you are using your phone for it; what is the make/model of phone?