I've never heard of melons or cucumbers producing suckers like tomato plants do. You might want to ask your question in the Vegetable gardening forum. Post a photo so they know what you're talking about. It might be the tendril the vine sends out to attach itself to something. I do know that it's best to get your plant into its final growing spot (container or ground) before it starts to vine. Cucurbits don't like their roots disturbed.

Here is a link that might be useful: Vegetable gardening forum