Generally speaking most companies project management has nothing to do with Customer Relationship Management (CRM) so there are no common transactions between those functions.
The same can be said about CRM and billing which is an accounting function.
Project management packages often generate resource usage and completion/milestone reports that can be used by accounting, but otherwise they are not accounting programs.
You may be able to tie functions together with some custom code and/or scripts to make report output from one function suitable as input to another function. The bottom line is that it is a do it yourself or outsourced programming task.
Microsoft application products have the capability to be customized through various versions of Visual Basic.
Though neither a full project management package nor a full accounting package Microsoft Office Outlook with Small Business Contact Manager may suit your purposes.
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