The HP ALM Synchronizer was designed for TFS 2008/2010 so the first issue you hit is that TF Services is today basically TFS 2013 (and is a moving goal post as it is updated so often). This means when you try to configure the TFS connection in HP ALM Synchronizer it fails because it cannot see any TFS client it supports. This is fairly simple to address, just install Visual Studio Team Explorer 2010 and patch it up to date so that it can connect to TF Service (you could go back to the 2008 and achieve the same if you really needed to)
Once you have a suitably old client you can progress to the point that it asks you for your TFS login credentials. HP ALM Synchronizer validates they are in the form DOMAIN\USER, this is a problem.
On TF Service you usually login with a LiveID, this is a non-starter in this case. However, you can configure alternative credentials, but these are in the form USER and PASSWORD. The string pattern verification on the credentials entry form in HP ALM Synchronizer does not accept them, it must have a domain and slash. I could not find any pattern at satisfies both TF Service and the HP ALM Synchronizer setup tool. So basically you are stuck.
So for my client we ended up moving to a TFS 2013 Basic Install on premises and at all worked fine, they could sync HP QC defects into TFS using the HP ALM Synchronizer, so they were happy.
However, is there a better solution? One might be to a commercial product such as Tasktop Sync, this is designed to provide synchronisation services between a whole range of ALM like products. I need to find out if that supports TF Service as yet?