How do I show a button only on my custom template page?

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It’s as simple as doing a conditional and check if you are on home.

What you are looking for is is_page_template( );. Wrap your button inside it as follows:

<?php if( is_page_template( 'your-template-slug.php' ) ){ ?>
    <button name="button" style=" ... " value="OK" type="button">HAVE A QUESTIONS</button>
<?php } ?>