There's no feature for that. UddeIM doesn't seam to support pre-filled PM messages either so I'm not sure if you can even construct a URL for that. You'd probably need to develop a custom CB plugin to have what you're wanting or use CB Auto Actions and create a PM or Email action with triggers set to None then link directly to it using a Custom HTML field for example.

You need to create an auto action and leave triggers as "None". Configure it however you want and use whatever action you want (e.g. Private Message). Now save it and under the Global tab you'll see its direct URL. It should be something like the following.

index.php?option=com_comprofiler&view=pluginclass&plugin=cbautoactions&action=action&actions=ACTION_ID

In a Custom HTML field you can link to that URL and include &users=[user_id] so the target user is the profile owner. Example as follows.

index.php?option=com_comprofiler&view=pluginclass&plugin=cbautoactions&action=action&actions=ACTION_ID&users=[user_id]

Your Private Message action could then be for example as follows.

From: [cb:userdata field="user_id" user="#me" /]
To: [user_id]
Subject: Hello!
Message: Testing!

Please ensure you're using latest nightly CB Auto Actions then retry.

To hide the link from a users self you can use CB Coniditionals to condition the user id of the profile against the viewing user id. Example as follows.

Field: Custom > Value
Value: [user_id]
Translate Value: No
Operator: Not Equal To
Value: [cb:userdata field="user_id" user="#me" /]
Translate Value: No
Mode: Hide