It comes from the core that way. So without hacking the core, you have 2 options.
Alternative 1
1. Locate the code from the core that generates that HTML here: https://github.com/q2a/question2answer/blob/7aa2774a87482e9143cf4da9aa650760a5cd57e6/qa-include/qa-page.php#L678
2. Copy the code and paste it in that function
3. Add whatever HTML you want
4. The resulting function should look like this:
public function logo() {
$newLogo = '<a href="'.qa_path_html('').'" class="qa-logo-link another-class" ... </a>;
$this->output(
'<div class="qa-logo">',
$newLogo,
'</div>'
);
}
Alternative 2
1. Just output your desired HTML
2. The resulting function should look like this (based on your example)
public function logo() {
$this->output(
'<div class="qa-logo">',
'<a class="qa-logo-link another-class" href="./">Website name</a>',
'</div>'
);
}
Alternative 3
1. You could also apply a string replacement that would search for qa-logo-link and replace it with qa-logo-link another-class