If your webserver is configured with Persistent Connections enabled (also known as keep-alive), then:
New content requested via AJAX after the original content had finished loading would require an additional connection but typically (depends on the browser implementation) this connection would be kept open for fetching assets required from the same domain until all requests have been responded to and then browser considered idle once more, at which time the browser will disconnect again.
Opening a different page will require a new connection unless the the link for it is clicked while an existing connection is still open, such as straight after page load or immediately following an AJAX request.
If persistent connections is not enabled on your webserver configuration then each request/response pair will use a separate connection. This can make most HTTP websites slower, but for HTTPS will be even more noticeable because of all the extra SSL/TLS handshaking required.
For a quick way to tell if your website has Persistent Connections enabled, test your website at www.webpagetest.org and then check the Keep Alive column in the table under the Performance Review tab.