I want to share a "contact me" link in posts on our forum.

For example on this forum, here's a link that opens a messenger page with myself as the recipient.   https://www.invisioncommunity.com/messenger/compose/?to=610431

How do I instead make a link that pops-up a floating messenger pane, rather than opens a new page? The way the messenger opens when you hover an avatar and click the "Message" button? 

Right click the element that you are trying to mimic, inspect it  and take a look at the code:

<a href="https://invisioncommunity.com/messenger/compose/?to=610431" data-ipsdialog="" data-ipsdialog-title="Compose New" data-ipsdialog-remotesubmit="" data-ipsdialog-flashmessage="Message Sent." data-ipsdialog-forcereload="" class="ipsFlex-flex:11 ipsButton ipsButton_light ipsButton_verySmall">Message</a>

There you go - that's your code; make necessary modifications add it via the Source button in the editor, or create a custom button in the editor to allow you to add it with the click of a button.

You can already do that without templates edit by using a free plugin: https://www.sosinvision.com.br/index.php?/file/250-user-options-in-postbit/

Great thank you both for your answers.