Clean the travertine well. Remove any dust or debris that may have gotten trapped inside the holes once the filling was removed. Spray stone cleaner onto the surface and buff the stone with a soft cloth until clean and dry.
Spread a thin layer of epoxy grout onto the surface of the travertine tile. Hold the float at a 45 degree angle to the tile and scrape it across the holes from various angles to fill them completely with the grout. Turn the float to 90 degrees and scrape away any excess grout from the areas surrounding the holes.
Let the grout dry for 10 minutes, then clean the stone around the holes with a grout sponge. Use the corners of the grout sponge to shape the edges of the new fill. Let the grout dry completely overnight.
Inspect the new fill. Some areas of fill may shrink during drying, causing it to slump or pull away from the edges. If this occurs, spread a new layer of grout over them until the fill is level with the rest of the tile.