No, default values of the same field are not customizable per template, and will remain the same on both templates regardless of if you change the value on one. However, you could use a workflow to update the value of a field on a template once it has been applied to a document.

For example, when Template 2 has been applied to a document, Workflow could automatically update the field value to the alternate default value you wanted to set. This would look seamless to the end user and achieve what you are looking to do.