The contact profile page is an Angular App located in \CMS\CMSScripts\CMSModules\CMS.ContactManagement\ContactProfile\
.
So in theory you can go and modify card.component.html (it is not recommended modifying Kentico code) :
<div class="header-actions-container">
<a ng-href="{{::$ctrl.contact.EditUrl}}" class="btn btn-primary">{$ contactmanagement.contact.editcontact $}</a>
<!-- put you link button here -->
</div>
<div class="head-section">
<h3 ng-bind="::$ctrl.contactCard.ContactName"></h3>
<cms-card-address ng-if="::$ctrl.showContactAddress" contact-card="::$ctrl.contactCard"/>
</div>
And then build a bundle, etc. But again you are touching the Kentico internal code. That might create problems with upgrades etc. Honestly I would contact Kentico support with this question.