Content API - parameter to find link url of a Link field
The Content API is a tricky thing to get the hang of, but try to do the following:

As long as your Rest Server endpoint path is "api" and the resource is "content" what you have above should technically work.  Make sure you also enabled the Link content type in the Content API configuration.

When you do query with /api/content?field_link=http://www.google.com you will only get the base fields returned, even if you filtered by http://www.google.com it won't show in the response you get.  You then have to send another request to /api/content/ and that will return every field for that given node.

Hopefully this helps a little bit, if anyone can fill in my holes I would appreciate it, I'm by no means an expert, just played around with the Content API and got it to work a little.

I did respond to your email with some private links to prove that it is possible.