There are several options, depending on what you would like to do.
OGC services
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 on images returned from WMS service. 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.
Add “additionalData” node with type “Date”
Click Save in Advanced parameters dialogue.
Click Save Layer and layer’s tab.
Catalog API
Make a request to Catalog API where you set distinct parameter to date along with setting bounding box, timespan and collection for which you want to get the acquisition dates (see example).