I believe that example below can push you in right direction.You can add simple and yet effective plugin L.EasyButton to get desired results.
I created little snippet for you, just to push you in the right direction, hope it helps:)
var map = L.map('map').setView([46.163613, 15.750947], 14);
mapLink = '<a href="http://openstreetmap.org">OpenStreetMap</a>';
L.tileLayer(
'http://{s}.tile.openstreetmap.org/{z}/{x}/{y}.png', {
attribution: '© ' + mapLink + ' Contributors',
maxZoom: 18,
}).addTo(map);
L.easyButton('fa-comment-o', function(btn, map) {
$('#myModal').modal('show');
}, 'Informacije').addTo(map);
#map {
width: 600px;
height: 400px;
}
<html>
<head>
<title>Custom Icons Tutorial - Leaflet</title>
<meta charset="utf-8" />
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<link rel="stylesheet" href="https://maxcdn.bootstrapcdn.com/bootstrap/3.3.7/css/bootstrap.min.css">
<link href="//maxcdn.bootstrapcdn.com/font-awesome/4.1.0/css/font-awesome.min.css" rel="stylesheet">
<link rel="stylesheet" href="https://unpkg.com/leaflet@1.0.3/dist/leaflet.css" />
<link href="https://unpkg.com/leaflet-easybutton@2.0.0/src/easy-button.css" rel="stylesheet">
<script src="https://ajax.googleapis.com/ajax/libs/jquery/3.2.1/jquery.min.js"></script>
<script src="https://unpkg.com/leaflet@1.0.3/dist/leaflet.js"></script>
<script src="https://unpkg.com/leaflet-easybutton@2.0.0/src/easy-button.js">
</script>
<script src="https://maxcdn.bootstrapcdn.com/bootstrap/3.3.7/js/bootstrap.min.js"></script>
</head>
<body>
<div id='map'></div>
<!-- Modal -->
<div class="modal fade" id="myModal" role="dialog">
<div class="modal-dialog">
<!-- Modal content-->
<div class="modal-content">
<div class="modal-header">
<button type="button" class="close" data-dismiss="modal">×</button>
<h4 class="modal-title">Modal Header</h4>
</div>
<div class="modal-body">
<p>Some text in the modal.</p>
</div>
<div class="modal-footer">
<button type="button" class="btn btn-default" data-dismiss="modal">Close</button>
</div>
</div>
</div>
</div>
</body>
</html>