Embedding from a site that uses a self-signed SSL certificate -

My site has an SSL certificate from a certified authority.

I have Piwik setup on one server in our office, and drag its tracking code on the view of each page.

I realize that if I want to avoid "unsolicited content" browser warnings, Piwik's JavaScript should work on SSL.

Can I use a self-signed certificate for this? Will the browser pick up on the fact that our site uses a self-signed certificate from any other site?

No, you can not use self-signed certificates for embedded code. If you do this then the browser will either complain or silently load unsafe content (unsafe because the identity can not be verified).