As in question, I'm looking for a way to connect any issue tracker with a development site hosted by Acquia Cloud.
I want to have a connection between git commits and issues.
Is this possible in Acquia?
I use the combo of BitBucket/JIRA. There are many people that like GitHub's issue tracker too. Acquia has tutorial on setting up either service as a 2nd remote repo. Either service will give you a way to track issues against your commits.