Ajax Filtering Pagination

I’d do it like this: (see the // comment here in the code)

// Wrap the AJAX call to `test_function` in a `function`.
function ajaxTestFunction( page_num ) {
    if (jQuery('#telegram_chk').is(":checked"))
    var telegram="";
    else
    var telegram="facebook";

    if (jQuery('#twitter_chk').is(":checked"))
    var twitter="";
    else 
    var twitter="twitter";

    if (jQuery('#eth_chk').is(":checked")) 
    var eth="";
    else 
    var eth="ethereum"; 

    if (jQuery('#xml_chk').is(":checked")) 
    var xml="";
    else 
    var xml="stellar";  


    var checkboxes = [telegram, twitter, eth, xml]

    jQuery.ajax({           

        url : rml_obj.ajax_url,
        type : 'post',
        data : {
            action : 'test_function',
            security : rml_obj.check_nonce,
            test_data : checkboxes,
            paged: page_num || 1
        },
        success : function( response ) {

            jQuery('#result').html(response);

        }
    });
}

// In the `change` callback, we call `ajaxTestFunction()`.
jQuery( '#content' ).on( 'change', '#telegram_chk, #twitter_chk, #eth_chk, #xml_chk', function( e ) {
    // Starts at page #1
    ajaxTestFunction();
});

// And add a listener/callback for the pagination clicks.
jQuery( '#result' ).on( 'click', '.pagination a', function( e ){
    e.preventDefault();

    var paged = /[\?&]paged=(\d+)/.test( this.href ) && RegExp.$1;

    ajaxTestFunction( paged );
});

And in the PHP test_function() function, capture paged like this:

$paged = $_POST['paged'];

..and not:

$paged = (get_query_var('paged')) ? get_query_var('paged') : 1;

Additional Note

Why don’t you utilize the paginate_links() function in WordPress? =)

The code starting from this line:

$pages =$q->max_num_pages;

..up until the } above this:

}    
else {
    echo "Sorry, we have not found any posts.";
}

..can be replaced with this:

$links = paginate_links([
    'base'    => '%_%',
    'format'  => '?paged=%#%',
    'total'   => $q->max_num_pages,
    'current' => $paged,
]);

if ( $links ) {
    echo '<div class="pagination">';
        echo $links;
    echo '</div>';
}

See https://codex.wordpress.org/Function_Reference/paginate_links and/or https://developer.wordpress.org/reference/functions/paginate_links/.