Yes, JIRA Mobile do support login to JIRA with google integration. To login, kindly follow the below steps:
Open your Cloud profile URL, ex: https://myjira.atlassian.net/secure/ViewProfile.jspa.
When you first logged in to JIRA with Google Apps integration, you would have been prompted to create an ‘Atlassian OnDemand’ password (It is this password that you need to use).