We wanted to use Intercom chat messenger on our Salesforce Community (Lightning Aura)

Hi @Madhura Kurdukar !

I’m not familiar with out Lightning Aura is set up. That said, if it allows you to add your own scripts to the HTML then you should be able to install the Intercom Messenger there just like you would on your own website or app. These two Help Center articles, discuss how to install the Messenger for Vistors/Leads and for Logged-in Users.