Link Tableau and R Shiny Dashboards

I am in the process of creating R Shiny Dashboard and am looking for a way to link with Tableau dashboard.

For example, if I have Tableau report A open, and I click on a button, can I transmit information through that click to let report B know that I want to drill down using a specific filter

Is "report B" the Shiny dashboard? If so, you'll want to look into whether/how you can get Tableau to open a URL with a specific query parameter, which you can then use to initialize your Shiny app.

Yes, Report B is Shiny Dashboard.