There are several options, depending on what you would like to do.
You can use a feature info request and point to some location. The response will contain a date field within it.
You can use the WFS service with the same parameters as used in your WMS request (e.g. date, cloud coverage). You will get a list of features representing scenes fitting the criteria.
You can configure the layer to show acquisition dates. Go to the layer of your choice and follow the steps:
Click the “Advanced” option in the Layer tab to enter advanced parameters dialogue.
(optional) You can turn on Help by clicking on “?” top right.