Hello Sumanta,
The XMPP API, that would come quite handy in your case, has been shut down at the end of October, as documented on the "XMPP API Shutdown"
page. There are
alternatives recommended on the "Alternatives to XMPP API" page.
After the XMPP API is turned off, you can obtain similar functionality by using any of a variety of 3rd party XMPP libraries in a language of your choice in an XMPP server running on a Compute Engine instance. For details on how to do this, take a look at the sample chat server Wikibot at GitHub.
Another example to examine, before implementing your live chat facility, is given on the "Multi-User Chat (MUC) Bot" page, and you may easily find a few other examples.
This group is meant for expressing general opinions, discussing trends and touching issues of general nature touching the app engine. In what concerns coding and programming details, it may be more effective to consult stackoverflow, where experienced programmers are within reach and ready to help.