You'll need to edit the modules/boonex/spy/classes/BxSpyProfilesActivity.php file. You can comment out what you don't want to show. For example:
//case 'request' :
//if($iSenderId != $iRecipientId) {
//$aRecipientInfo = $this -> _getSenderInfo($iSenderId);
//$sRecipientNickName = $aRecipientInfo['NickName'];
//$sRecipientProfileLink = $aRecipientInfo['Link'];
//$aParams = array(
//'lang_key' => '_bx_spy_profile_friend_request',
//'params' => array(
//'sender_p_link' => $sProfileLink,
//'sender_p_nick' => $sNickName,
//'recipient_p_link' => $sRecipientProfileLink,
//'recipient_p_nick' => $sRecipientNickName,
//),
//);
//}
//break;
//case 'accept' :
//if($iSenderId != $iRecipientId) {
//$aSenderInfo = $this -> _getSenderInfo($iSenderId);
//$sSenderNickName = $aSenderInfo['NickName'];
//$sSenderProfileLink = $aSenderInfo['Link'];
//$aParams = array(
//'lang_key' => '_bx_spy_profile_friend_accept',
//'params' => array(
//'sender_p_link' => $sProfileLink,
//'sender_p_nick' => $sNickName,
//'recipient_p_link' => $sSenderProfileLink,
//'recipient_p_nick' => $sSenderNickName,
//),
//);
//}
//break;