Is there any exist services or way to add 1km x 1km UTM grid onto online map service such as Google Maps or OpenStreetMap?
there are several ways you could achieve this
1) use some sort of online GIS platform like GIS Cloud or MangoMap, where you could upload your 1km x 1km grid and have OSM or google as the base map and then save the project and make it publicly accessible. And then send the URL of that page to people who want to see it. Or both of these services allow you to embed an iframe into your own webpage which you could host on your web server.
This is straight forward and would only need you to have the 1km grid.
2) Write your own leaflet or OpenLayers html page with the base map as OSM or Google and then overlay the 1km grid as a vector overlay. You could use KML or GeoJSON as the vector overlay
How to create the 1km grid:
One way is to use QGIS use Research Tools> Vector Grid to create the 1km grid.
WARNING: depending on the area you want the 1km grid to cover this file could be huge and if loading as a vector layer (option 2 above) then the performance would be terrible.
if this is the case you could use TopoJSON to reduce the amount of geometries of your 1km grid. For more details on TopoJSON see here https://github.com/mbostock/topojson/ and Mike's tutorial here http://bost.ocks.org/mike/map/
Hope that helps